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Maths-

General

Easy

Question

Let f be twice differentiable real valued function satisfying $f to the power of ´ end exponent left parenthesis x right parenthesis not equal to 0 comma f left parenthesis x right parenthesis plus f to the power of ´ ´ end exponent left parenthesis x right parenthesis equals negative x g left parenthesis x right parenthesis f to the power of ´ end exponent left parenthesis x right parenthesis$ where $g left parenthesis x right parenthesis greater than 0 for all x greater than 0 text .If end text f left parenthesis 0 right parenthesis equals negative 3 text end text$ and $f to the power of ´ end exponent left parenthesis 0 right parenthesis equals 4 text end text$ then
Statement -1: $vertical line f left parenthesis x right parenthesis vertical line less or equal than 5 for all x greater than 0$
Statement -2: $left parenthesis f left parenthesis x right parenthesis right parenthesis to the power of 2 end exponent plus open parentheses f to the power of ´ end exponent left parenthesis x right parenthesis close parentheses to the power of 2 end exponent$ is decreasing $for all x greater than 0.$

Statement-1 is True,Statement-2 is True;Statement-2 is a correct explanation for Statement-1
Statement -1 is True, Statement -2 is True;Statement-2 is NOT a correct explanation for Statement-1
Statement -1 is True, Statement -2 is False
Statement -1 is False, Statement -2 is True.

The correct answer is: Statement-1 is True,Statement-2 is True;Statement-2 is a correct explanation for Statement-1

$fraction numerator d left parenthesis f left parenthesis x right parenthesis right parenthesis to the power of 2 end exponent plus open parentheses f to the power of ´ end exponent left parenthesis x right parenthesis close parentheses to the power of 2 end exponent over denominator d x end fraction equals 2 f to the power of ´ end exponent left parenthesis x right parenthesis open parentheses f left parenthesis x right parenthesis plus f to the power of ´ ´ end exponent left parenthesis x right parenthesis close parentheses equals negative 2 x g left parenthesis x right parenthesis open parentheses f to the power of ´ end exponent left parenthesis x right parenthesis close parentheses to the power of 2 end exponent less than 0 for all x greater than 0$
$table row cell therefore left parenthesis f left parenthesis x right parenthesis right parenthesis to the power of 2 end exponent less than 25 minus open parentheses f to the power of ´ end exponent left parenthesis x right parenthesis close parentheses to the power of 2 end exponent less than 25 end cell row cell therefore vertical line f left parenthesis x right parenthesis vertical line less than 5. end cell end table$

Related Questions to study

Let $I equals stretchy integral subscript 0 end subscript superscript 1 end superscript fraction numerator sin invisible function application x over denominator square root of x end fraction d x text and end text J equals stretchy integral subscript 0 end subscript superscript 1 end superscript fraction numerator cos invisible function application x over denominator square root of x end fraction d x text . end text$ Then which one of the following is true ?

Let $I equals stretchy integral subscript 0 end subscript superscript 1 end superscript fraction numerator sin invisible function application x over denominator square root of x end fraction d x text and end text J equals stretchy integral subscript 0 end subscript superscript 1 end superscript fraction numerator cos invisible function application x over denominator square root of x end fraction d x text . end text$ Then which one of the following is true ?

$text Iff end text text . end text minus y minus e to the power of y end exponent comma g comma negative y minus y comma y greater than 0 text end text$ andE.. $t minus stretchy integral subscript 0 end subscript superscript t end superscript f left parenthesis t minus y right parenthesis g left parenthesis y right parenthesis d y text , end text$ then-

$text Iff end text text . end text minus y minus e to the power of y end exponent comma g comma negative y minus y comma y greater than 0 text end text$ andE.. $t minus stretchy integral subscript 0 end subscript superscript t end superscript f left parenthesis t minus y right parenthesis g left parenthesis y right parenthesis d y text , end text$ then-

If for a non-zero x, af $x plus b f open parentheses fraction numerator 1 over denominator x end fraction close parentheses equals fraction numerator 1 over denominator x end fraction minus 5 text , end text$ where $a to the power of not equal to end exponent b text , end text$ then $stretchy integral subscript 1 end subscript superscript 2 end superscript$ f(x)dx=

If for a non-zero x, af $x plus b f open parentheses fraction numerator 1 over denominator x end fraction close parentheses equals fraction numerator 1 over denominator x end fraction minus 5 text , end text$ where $a to the power of not equal to end exponent b text , end text$ then $stretchy integral subscript 1 end subscript superscript 2 end superscript$ f(x)dx=

parallel

Which of the following is correct?

Which of the following is correct?

chemistry-General

In the thermal power stations, electricity is produced by producing heat energy from the burning of coal. This is an example of ..

In the thermal power stations, electricity is produced by producing heat energy from the burning of coal. This is an example of ..

chemistry-General

Burning of a candle is an example of ..

Burning of a candle is an example of ..

chemistry-General

parallel

A bob attached to a string is held horizontal and released. The tension and vertical distance from point of suspension can be represented by.

A bob attached to a string is held horizontal and released. The tension and vertical distance from point of suspension can be represented by.

physics-General

A ball whose size is slightly smaller than width of the tube of radius 2.5 m is projected from bottommost point of a smooth tube fixed in a vertical plane with velocity of 10 m/s. If N₁ and N₂ are the normal reactions exerted by inner side and outer side of the tube on the ball

A ball whose size is slightly smaller than width of the tube of radius 2.5 m is projected from bottommost point of a smooth tube fixed in a vertical plane with velocity of 10 m/s. If N₁ and N₂ are the normal reactions exerted by inner side and outer side of the tube on the ball

physics-General

A conical pendulum is moving in a circle with angular velocity was shown. If tension in the string is T, which of following equations are correct ?

A conical pendulum is moving in a circle with angular velocity was shown. If tension in the string is T, which of following equations are correct ?

physics-General

parallel

A pendulum bob is swinging in a vertical plane such that its angular amplitude is less than 900 . At its highest point, the string is cut. Which trajectory is possible for the bob afterwards.

A pendulum bob is swinging in a vertical plane such that its angular amplitude is less than 900 . At its highest point, the string is cut. Which trajectory is possible for the bob afterwards.

physics-General

Block of 1 kg is initially in equilibrium and is hanging by two identical springs A and B as shown in figures. If spring A is cut from lower point at t=0 then, find acceleration of block in ms–2 at t = 0.

Block of 1 kg is initially in equilibrium and is hanging by two identical springs A and B as shown in figures. If spring A is cut from lower point at t=0 then, find acceleration of block in ms–2 at t = 0.

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Find the acceleration of 3 kg mass when acceleration of 2 kg mass is 2 ms^–2 as shown in figure.

Find the acceleration of 3 kg mass when acceleration of 2 kg mass is 2 ms^–2 as shown in figure.

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A rope of mass 5 kg is moving vertically in vertical position with an upwards force of 100 N acting at the upper end and a downwards force of 70 N acting at the lower end. The tension at midpoint of the rope is

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physics-General

A ball is hit by a batsman at an angle of 37° as shown in figure. The man standing at P should run at what minimum velocity so that he catches the ball before it strikes the ground. Assume that height of man is negligible in comparison to maximum height of projectile.

A ball is hit by a batsman at an angle of 37° as shown in figure. The man standing at P should run at what minimum velocity so that he catches the ball before it strikes the ground. Assume that height of man is negligible in comparison to maximum height of projectile.

physics-General

Average velocity of a particle is projectile motion between its starting point and the highest point of its trajectory is : (projection speed = u, angle of projection from horizontal= $theta$ )

Average velocity of a particle is projectile motion between its starting point and the highest point of its trajectory is : (projection speed = u, angle of projection from horizontal= $theta$ )

physics-General

parallel

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