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Question

At what points of curve $y equals fraction numerator 2 over denominator 3 end fraction x to the power of 3 end exponent plus fraction numerator 1 over denominator 2 end fraction x to the power of 2 end exponent$ , the tangent makes the equal angle with the axis?

$(\frac{1}{2}, \frac{5}{24})$ and $(- 1, - \frac{1}{6})$
$(\frac{1}{2}, \frac{4}{9})$ and $(- 1, 0)$
$(\frac{1}{3}, \frac{1}{7})$ and $(- 3, \frac{1}{2})$
$(\frac{1}{3}, \frac{4}{47})$ and $(- 1, - \frac{1}{3})$

The correct answer is: $open parentheses fraction numerator 1 over denominator 2 end fraction comma fraction numerator 5 over denominator 24 end fraction close parentheses$ and $open parentheses negative 1 comma blank minus fraction numerator 1 over denominator 6 end fraction close parentheses$

$y equals fraction numerator 2 over denominator 3 end fraction x to the power of 3 end exponent plus fraction numerator 1 over denominator 2 end fraction x to the power of 2 end exponent$
$therefore fraction numerator d y over denominator d x end fraction equals fraction numerator 2 over denominator 3 end fraction 3 x to the power of 2 end exponent plus fraction numerator 1 over denominator 2 end fraction 2 x equals 2 x to the power of 2 end exponent plus blank x$
Since the tangent makes equal angles with the axes
$rightwards double arrow fraction numerator d y over denominator d x end fraction equals plus-or-minus 1$
$rightwards double arrow 2 x to the power of 2 end exponent plus x equals plus-or-minus 1$
$rightwards double arrow 2 x to the power of 2 end exponent plus x minus 1 equals 0 blank left parenthesis 2 x to the power of 2 end exponent plus blank x plus 1 equals 0 blank h a s blank n o blank r e a l blank r o o t s right parenthesis$
$rightwards double arrow open parentheses 2 x minus 1 close parentheses open parentheses x plus 1 close parentheses equals blank 0$
$rightwards double arrow x equals fraction numerator 1 over denominator 2 end fraction blank o r blank x equals negative 1$

Related Questions to study

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The fuel charges for running a train are proportional to the square of the speed generated in km per hour, and the cost is Rs. 48 at 16 km per hour. If the fixed charges amount to Rs. 300 per hour, the most economical speed is

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$increment subscript 1 end subscript equals open vertical bar table row cell y to the power of 5 end exponent z to the power of 6 end exponent left parenthesis z to the power of 3 end exponent minus y to the power of 3 end exponent right parenthesis end cell cell x to the power of 4 end exponent z to the power of 6 end exponent left parenthesis x to the power of 3 end exponent minus z to the power of 3 end exponent right parenthesis end cell cell x to the power of 4 end exponent y to the power of 5 end exponent left parenthesis y to the power of 3 end exponent minus x to the power of 3 end exponent right parenthesis end cell row cell y to the power of 2 end exponent z to the power of 3 end exponent left parenthesis y to the power of 6 end exponent minus z to the power of 6 end exponent right parenthesis end cell cell x z to the power of 3 end exponent left parenthesis z to the power of 6 end exponent minus x to the power of 6 end exponent right parenthesis end cell cell x y to the power of 2 end exponent left parenthesis x to the power of 6 end exponent minus y to the power of 6 end exponent right parenthesis end cell row cell y to the power of 2 end exponent z to the power of 3 end exponent left parenthesis z to the power of 3 end exponent minus y to the power of 3 end exponent right parenthesis end cell cell x z to the power of 3 end exponent left parenthesis x to the power of 3 end exponent minus z to the power of 3 end exponent right parenthesis end cell cell x y to the power of 2 end exponent left parenthesis y to the power of 3 end exponent minus x to the power of 3 end exponent right parenthesis end cell end table close vertical bar$ 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

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