The local maximum value of the function f(x)=b^2/a^4(a^2x^2-x^4), a,b>0 is equal to: Option 1: b^2/4 Option 2: b^2/2 Option 3: 2b^2 Option 4: 4b^2

Answer: b^2/4. Solution: f'(x) = b^2/a^4(2a^2x-4x^3) = 0 gives x = 0 or x^2 = a^2/2. The second derivative is negative at x^2 = a^2/2, so these are maxima. There a^2x^2-x^4 = a^4/2-a^4/4 = a^4/4, giving f = b^2/4.

Calculus

Application of Derivatives

Medium

core

The local maximum value of the function $f(x)=\dfrac{b^2}{a^4}(a^2x^2-x^4), a,b>0$ is equal to:

$\dfrac{b^2}{4}$

$\dfrac{b^2}{2}$

$2b^2$

$4b^2$

✅ Correct Option: 1

$f'(x) = \dfrac{b^2}{a^4}(2a^2x-4x^3) = 0$ gives $x = 0$ or $x^2 = \dfrac{a^2}{2}$ .

The second derivative is negative at $x^2 = \dfrac{a^2}{2}$ , so these are maxima.

There $a^2x^2-x^4 = \dfrac{a^4}{2}-\dfrac{a^4}{4} = \dfrac{a^4}{4}$ , giving $f = \dfrac{b^2}{4}$ .

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The function $f(x)=\cot^{-1}x+x$ increases in the interval:

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If $f(x) = 2x^3 - 15x^2 + 36x + 1$ , $x \in [1, 5]$ , then the absolute minimum value of $f(x)$ is:

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Match List-I with List-II

List-I	List-II
(A) Point of minima of $f(x) = \vert x+1\vert$	(I) 1
(B) Minimum value of $f(x) = \vert x\vert$	(II) -1
(C) Maximum value of $f(x) = 1 - x^2$	(III) 2
(D) Minimum value of $f(x) = 2 + \sin^2 x$	(IV) 0

Choose the correct answer from the options given below:

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Rate of change of area of a circle with respect to radius

(A) When r = 2 cm is $4\pi$ cm²/cm.

(B) When r = 6 cm is $12\pi$ cm²/cm.

(D) When r = 16 cm is $8\pi$ cm²/cm.

Choose the correct answer from the options given below:

2023: 22 May Shift 3

If the rate of change of area of a circle is equal to the rate of change of its diameter, then its radius is equal to :

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The function $f(x) = 4x^3 - 7x^2$ has point(s) of local minima at

The local maximum value of the function f(x)=b2a4(a2x2−x4),a,b>0f(x)=\dfrac{b^2}{a^4}(a^2x^2-x^4), a,b>0f(x)=a4b2​(a2x2−x4),a,b>0 is equal to:

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The local maximum value of the function $f(x)=\dfrac{b^2}{a^4}(a^2x^2-x^4), a,b>0$ is equal to: