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Question

Suppose that $f$ is a polynomial of degree 3 and that $f to the power of ´ ´ end exponent left parenthesis x right parenthesis not equal to 0$ at any of the stationary point. Then

f.has exactly one stationary point
f must have no stationary point
f must have exactly two stationary points
f has either zero or two stationary point

The correct answer is: f has either zero or two stationary point

$y equals fraction numerator log invisible function application x over denominator x end fraction$
$rightwards double arrow fraction numerator d y over denominator d x end fraction equals negative fraction numerator 1 over denominator x to the power of 2 end exponent end fraction log invisible function application x plus fraction numerator 1 over denominator x end fraction fraction numerator 1 over denominator x end fraction$
$equals fraction numerator 1 over denominator x to the power of 2 end exponent end fraction open parentheses 1 minus log invisible function application x close parentheses equals 0$
$fraction numerator d y over denominator d x end fraction equals 0 rightwards double arrow log invisible function application x equals 1 o r blank x equals e$
For $x less than e rightwards double arrow log invisible function application x less than 1$
and $x greater than e rightwards double arrow log invisible function application x greater than 1$
At $x equals e comma fraction numerator d y over denominator d x end fraction$ changes sign from +ve to $–$ ve and hence $y$ is maximum at $x equals e$ and its value is
$fraction numerator log invisible function application e over denominator e end fraction equals e to the power of negative 1 end exponent$
Also, $f to the power of ´ ´ end exponent open parentheses x close parentheses equals 2 m left parenthesis x minus alpha right parenthesis$ , in which case $f to the power of ´ ´ end exponent open parentheses alpha close parentheses equals 0$ . But we are told that the 2nd derivative is non-zero at critical point. Hence, there must be either 0 or 2 critical points

Related Questions to study

Let $f open parentheses x close parentheses equals cos invisible function application pi x plus 10 x plus 3 x to the power of 2 end exponent plus x to the power of 3 end exponent comma blank minus 2 less or equal than x less or equal than 3$ . The absolute minimum value of $f left parenthesis x right parenthesis$ is

Let $f open parentheses x close parentheses equals cos invisible function application pi x plus 10 x plus 3 x to the power of 2 end exponent plus x to the power of 3 end exponent comma blank minus 2 less or equal than x less or equal than 3$ . The absolute minimum value of $f left parenthesis x right parenthesis$ is

Let $f open parentheses x close parentheses equals open curly brackets table row cell x plus 2 comma blank minus 1 less or equal than x less than 0 end cell row cell 1 comma blank x equals 0 blank end cell row cell fraction numerator x over denominator 2 end fraction comma blank 0 less than x less or equal than 1 end cell end table close$ Then on $left square bracket negative 1 comma blank 1 right square bracket$ , this function has

Let $f open parentheses x close parentheses equals open curly brackets table row cell x plus 2 comma blank minus 1 less or equal than x less than 0 end cell row cell 1 comma blank x equals 0 blank end cell row cell fraction numerator x over denominator 2 end fraction comma blank 0 less than x less or equal than 1 end cell end table close$ Then on $left square bracket negative 1 comma blank 1 right square bracket$ , this function has

If $ϕ left parenthesis x right parenthesis$ is a polynomial function and $ϕ to the power of ´ end exponent open parentheses x close parentheses greater than ϕ open parentheses x close parentheses comma blank for all blank x greater or equal than 1$ and $ϕ open parentheses 1 close parentheses equals 0$ , then

If $ϕ left parenthesis x right parenthesis$ is a polynomial function and $ϕ to the power of ´ end exponent open parentheses x close parentheses greater than ϕ open parentheses x close parentheses comma blank for all blank x greater or equal than 1$ and $ϕ open parentheses 1 close parentheses equals 0$ , then

parallel

If $f open parentheses x close parentheses equals x plus sin invisible function application x semicolon g open parentheses x close parentheses equals e to the power of negative x end exponent semicolon u equals square root of c plus 1 end root minus square root of c semicolon v equals square root of c minus square root of c minus 1 end root semicolon left parenthesis c greater than 1 right parenthesis$ , then

If $f open parentheses x close parentheses equals x plus sin invisible function application x semicolon g open parentheses x close parentheses equals e to the power of negative x end exponent semicolon u equals square root of c plus 1 end root minus square root of c semicolon v equals square root of c minus square root of c minus 1 end root semicolon left parenthesis c greater than 1 right parenthesis$ , then

The value of $c$ in Lagrange’s theorem for the function $f open parentheses x close parentheses equals log invisible function application sin invisible function application x$ in the interval $left square bracket divided by 6 comma blank 5 divided by 6 right square bracket$

The value of $c$ in Lagrange’s theorem for the function $f open parentheses x close parentheses equals log invisible function application sin invisible function application x$ in the interval $left square bracket divided by 6 comma blank 5 divided by 6 right square bracket$

Let $f left parenthesis x right parenthesis$ be a function such that $f to the power of ´ end exponent open parentheses x close parentheses equals log subscript 1 divided by 3 end subscript invisible function application left square bracket log subscript 3 end subscript invisible function application left parenthesis sin invisible function application x plus a right parenthesis right square bracket$ . If $f left parenthesis x right parenthesis$ is decreasing for all real values of $x$ , then

Let $f left parenthesis x right parenthesis$ be a function such that $f to the power of ´ end exponent open parentheses x close parentheses equals log subscript 1 divided by 3 end subscript invisible function application left square bracket log subscript 3 end subscript invisible function application left parenthesis sin invisible function application x plus a right parenthesis right square bracket$ . If $f left parenthesis x right parenthesis$ is decreasing for all real values of $x$ , then

parallel

The volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle $30 degree$ is

The volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle $30 degree$ is

Consider the system of equation $x plus y plus z equals 6 comma blank x plus 2 y plus 3 z equals 10$ and $x plus 2 y plus lambda z equals mu$
Statement 1 If the system has infinite number of solutions, then $mu equals 10$
Statement 2: The determinant $open vertical bar table row 1 1 6 row 1 2 10 row 1 2 mu end table close vertical bar equals 0$ for $mu equals 10$

Consider the system of equation $x plus y plus z equals 6 comma blank x plus 2 y plus 3 z equals 10$ and $x plus 2 y plus lambda z equals mu$
Statement 1 If the system has infinite number of solutions, then $mu equals 10$
Statement 2: The determinant $open vertical bar table row 1 1 6 row 1 2 10 row 1 2 mu end table close vertical bar equals 0$ for $mu equals 10$

If $p q r not equal to 0$ and the system of equations $blank open parentheses p plus a close parentheses x plus b y plus c z equals 0 a x plus open parentheses q plus b close parentheses y plus c z equals 0$ $a x plus b y plus open parentheses r plus c close parentheses z equals 0$ Has a non-trivial solution, then value of $fraction numerator a over denominator p end fraction plus fraction numerator b over denominator q end fraction plus fraction numerator c over denominator r end fraction$ is

If $p q r not equal to 0$ and the system of equations $blank open parentheses p plus a close parentheses x plus b y plus c z equals 0 a x plus open parentheses q plus b close parentheses y plus c z equals 0$ $a x plus b y plus open parentheses r plus c close parentheses z equals 0$ Has a non-trivial solution, then value of $fraction numerator a over denominator p end fraction plus fraction numerator b over denominator q end fraction plus fraction numerator c over denominator r end fraction$ is

parallel

If $a comma blank b comma blank c$ are non-zeros, then the system of equations $open parentheses alpha plus a close parentheses x plus alpha y plus alpha z equals 0 comma blank alpha x plus open parentheses alpha plus b close parentheses y plus alpha z equals 0 comma alpha x plus alpha y plus open parentheses alpha plus c close parentheses z equals 0$ has a non-trivial solution if

If $a comma blank b comma blank c$ are non-zeros, then the system of equations $open parentheses alpha plus a close parentheses x plus alpha y plus alpha z equals 0 comma blank alpha x plus open parentheses alpha plus b close parentheses y plus alpha z equals 0 comma alpha x plus alpha y plus open parentheses alpha plus c close parentheses z equals 0$ has a non-trivial solution if

$increment subscript 1 end subscript equals open vertical bar table row cell y to the power of 5 end exponent z to the power of 6 end exponent left parenthesis z to the power of 3 end exponent minus y to the power of 3 end exponent right parenthesis end cell cell x to the power of 4 end exponent z to the power of 6 end exponent left parenthesis x to the power of 3 end exponent minus z to the power of 3 end exponent right parenthesis end cell cell x to the power of 4 end exponent y to the power of 5 end exponent left parenthesis y to the power of 3 end exponent minus x to the power of 3 end exponent right parenthesis end cell row cell y to the power of 2 end exponent z to the power of 3 end exponent left parenthesis y to the power of 6 end exponent minus z to the power of 6 end exponent right parenthesis end cell cell x z to the power of 3 end exponent left parenthesis z to the power of 6 end exponent minus x to the power of 6 end exponent right parenthesis end cell cell x y to the power of 2 end exponent left parenthesis x to the power of 6 end exponent minus y to the power of 6 end exponent right parenthesis end cell row cell y to the power of 2 end exponent z to the power of 3 end exponent left parenthesis z to the power of 3 end exponent minus y to the power of 3 end exponent right parenthesis end cell cell x z to the power of 3 end exponent left parenthesis x to the power of 3 end exponent minus z to the power of 3 end exponent right parenthesis end cell cell x y to the power of 2 end exponent left parenthesis y to the power of 3 end exponent minus x to the power of 3 end exponent right parenthesis end cell end table close vertical bar$ and $increment subscript 2 end subscript equals open vertical bar table row x cell y to the power of 2 end exponent end cell cell z to the power of 3 end exponent end cell row cell x to the power of 4 end exponent end cell cell y to the power of 5 end exponent end cell cell z to the power of 6 end exponent end cell row cell x to the power of 7 end exponent end cell cell y to the power of 8 end exponent end cell cell z to the power of 9 end exponent end cell end table close vertical bar$ . Then $increment subscript 1 end subscript increment subscript 2 end subscript$ is equal to

$increment subscript 1 end subscript equals open vertical bar table row cell y to the power of 5 end exponent z to the power of 6 end exponent left parenthesis z to the power of 3 end exponent minus y to the power of 3 end exponent right parenthesis end cell cell x to the power of 4 end exponent z to the power of 6 end exponent left parenthesis x to the power of 3 end exponent minus z to the power of 3 end exponent right parenthesis end cell cell x to the power of 4 end exponent y to the power of 5 end exponent left parenthesis y to the power of 3 end exponent minus x to the power of 3 end exponent right parenthesis end cell row cell y to the power of 2 end exponent z to the power of 3 end exponent left parenthesis y to the power of 6 end exponent minus z to the power of 6 end exponent right parenthesis end cell cell x z to the power of 3 end exponent left parenthesis z to the power of 6 end exponent minus x to the power of 6 end exponent right parenthesis end cell cell x y to the power of 2 end exponent left parenthesis x to the power of 6 end exponent minus y to the power of 6 end exponent right parenthesis end cell row cell y to the power of 2 end exponent z to the power of 3 end exponent left parenthesis z to the power of 3 end exponent minus y to the power of 3 end exponent right parenthesis end cell cell x z to the power of 3 end exponent left parenthesis x to the power of 3 end exponent minus z to the power of 3 end exponent right parenthesis end cell cell x y to the power of 2 end exponent left parenthesis y to the power of 3 end exponent minus x to the power of 3 end exponent right parenthesis end cell end table close vertical bar$ and $increment subscript 2 end subscript equals open vertical bar table row x cell y to the power of 2 end exponent end cell cell z to the power of 3 end exponent end cell row cell x to the power of 4 end exponent end cell cell y to the power of 5 end exponent end cell cell z to the power of 6 end exponent end cell row cell x to the power of 7 end exponent end cell cell y to the power of 8 end exponent end cell cell z to the power of 9 end exponent end cell end table close vertical bar$ . Then $increment subscript 1 end subscript increment subscript 2 end subscript$ is equal to

The system of equations $a x minus y minus z equals alpha minus 1$ $x minus alpha y minus z equals alpha minus 1$ $x minus y minus alpha z equals alpha minus 1$ Has no solution if $alpha$ is

The system of equations $a x minus y minus z equals alpha minus 1$ $x minus alpha y minus z equals alpha minus 1$ $x minus y minus alpha z equals alpha minus 1$ Has no solution if $alpha$ is

parallel

The value of the determinant $open vertical bar table row 1 1 1 row cell blank to the power of m end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 2 end exponent C subscript 1 end subscript end cell row cell blank to the power of m end exponent C subscript 2 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 2 end subscript end cell cell blank to the power of m plus 2 end exponent C subscript 2 end subscript end cell end table close vertical bar$ is equal to

The value of the determinant $open vertical bar table row 1 1 1 row cell blank to the power of m end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 2 end exponent C subscript 1 end subscript end cell row cell blank to the power of m end exponent C subscript 2 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 2 end subscript end cell cell blank to the power of m plus 2 end exponent C subscript 2 end subscript end cell end table close vertical bar$ is equal to

If $alpha comma blank beta comma blank gamma$ are the angles of a triangle and the system of equations $cos invisible function application open parentheses alpha minus beta close parentheses x plus cos invisible function application open parentheses beta minus gamma close parentheses y plus cos invisible function application open parentheses gamma minus alpha close parentheses z equals 0 cos invisible function application open parentheses alpha plus beta close parentheses x plus cos invisible function application open parentheses beta plus gamma close parentheses y plus cos invisible function application open parentheses gamma plus alpha close parentheses z equals 0 sin invisible function application open parentheses alpha plus beta close parentheses x plus sin invisible function application open parentheses beta plus gamma close parentheses y plus sin invisible function application open parentheses gamma plus alpha close parentheses z equals 0$ Has non-trivial solutions, then triangle is necessarily

If $alpha comma blank beta comma blank gamma$ are the angles of a triangle and the system of equations $cos invisible function application open parentheses alpha minus beta close parentheses x plus cos invisible function application open parentheses beta minus gamma close parentheses y plus cos invisible function application open parentheses gamma minus alpha close parentheses z equals 0 cos invisible function application open parentheses alpha plus beta close parentheses x plus cos invisible function application open parentheses beta plus gamma close parentheses y plus cos invisible function application open parentheses gamma plus alpha close parentheses z equals 0 sin invisible function application open parentheses alpha plus beta close parentheses x plus sin invisible function application open parentheses beta plus gamma close parentheses y plus sin invisible function application open parentheses gamma plus alpha close parentheses z equals 0$ Has non-trivial solutions, then triangle is necessarily

If $x not equal to y not equal to z$ and $open vertical bar table row x cell x to the power of 2 end exponent end cell cell 1 plus x to the power of 3 end exponent end cell row y cell y to the power of 2 end exponent end cell cell 1 plus y to the power of 3 end exponent end cell row z cell z to the power of 2 end exponent end cell cell 1 plus z to the power of 3 end exponent end cell end table close vertical bar equals 0$ , then the value of $x y z$ is

If $x not equal to y not equal to z$ and $open vertical bar table row x cell x to the power of 2 end exponent end cell cell 1 plus x to the power of 3 end exponent end cell row y cell y to the power of 2 end exponent end cell cell 1 plus y to the power of 3 end exponent end cell row z cell z to the power of 2 end exponent end cell cell 1 plus z to the power of 3 end exponent end cell end table close vertical bar equals 0$ , then the value of $x y z$ is

parallel

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