Calculus

Application of Derivatives

Medium

core

For $f(x) = x^3 - 6x^2 + 9x + 15$ , which of the following statements are TRUE?

(A) $f'(x) = 3x^2 - 12x + 15$ .

(B) The critical points are 3 and 1.

(C) $x = 1$ is a point of local maxima.

(D) Local minimum value is 19.

Choose the correct answer from the options given below:

(A) and (C) only

(A) and (D) only

(B) and (C) only

(B) and (D) only

✅ Correct Option: 3

$f'(x) = 3x^2 - 12x + 9$ , so (A) is false. Its roots are $x = 1, 3$ , so (B) is true. Since $f''(1) = -6 < 0$ , $x = 1$ is a local maximum, so (C) is true. The local minimum is $f(3) = 15$ , not 19, so (D) is false.

2026: 19 May Shift 1

The absolute maximum value of the function $f(x) = 2x^3 - 15x^2 + 36x + 7$ on the interval [1,5] is

2025: 19 May Shift 1

Function $f(x) = x^3 - 3x + 3$ is

(A) Increasing in the interval $(-1, 1)$

(B) Increasing in the interval $(1, \infty)$

(D) Increasing in the interval $(-\infty, -1) \cup (1, \infty)$

Choose the correct answer from the options given below:

2025: 2 June Shift 1

The function $f(x) = 2\log_e(x-2) - x^2 + 4x + 1, (x > 2)$ is increasing on the interval:

2026: 18 May Shift 1

A solid sphere of gold of radius 5 cm, is being melted in a furnace such that the radius is decreasing uniformly. When the radius is 1cm, the rate at which surface area (A) is decreasing w.r.t. time ( $t$ ) is equal to

( $\frac{dv}{dt}$ = rate at which volume is decreasing w.r.t. $t$ )

2023: 15 June Shift 2

The rate of change of the area of a circular disc with respect to its circumference when radius is 3 is :

2022: 30 Aug Shift 1

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

Solution

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