Calculus

Application of Derivatives

Easy

common

Consider the function $f(x) = -2x^3 + 3x^2 + 36x - 15$ , then which of the following are TRUE?

A. $f(x)$ increases on $(-2, 3)$

B. $f(x)$ decreases on $(-\infty, -2) \cup (3, \infty)$

C. $f(x)$ increases on $(-\infty, -2) \cup (3, \infty)$

D. $f(x)$ neither increases nor decreases on $(-2, 3)$

Choose the correct answer from the options given below:

A and B only

A and D only

B and D only

C and D only

✅ Correct Option: 1

$f'(x) = -6x^2 + 6x + 36 = -6(x-3)(x+2)$ .

This is positive exactly when $-2 < x < 3$ , so $f$ increases on $(-2,3)$ and decreases on $(-\infty,-2)$ and $(3,\infty)$ . Hence A and B are true, C and D are false.

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Match List-I with List-II

List-I	List-II
(a) If $x = t^2$ and $y = t^3$ , then $\frac{d^2y}{dx^2}$ at $t = 1$	(i) $-2$
(b) If $f(x) = \sqrt{x} + 1$ , then $f''(1)$	(ii) $-1$
(c) The minimum value of $f(x) = 9x^2 + 12x + 2$ is	(iii) $\frac{3}{4}$
(d) The point of inflexion of the function $f(x) = (x-2)^4 (x+1)^3$ is	(iv) $-\frac{1}{4}$

Choose the correct answer from the options given below

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Based on above information answer the following question :

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Solution

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