Calculus

Application of Derivatives

Medium

common

For the function $f(x)=2x^3-21x^2+36x-100$ ;

Match the LIST-I with LIST-II

LIST-I LIST-II
A. Point of local maxima is I. -208
B. Point of local minima is II. -83
C. Local maximum value is III. 6
D. Local minimum value is IV. 1

Choose the correct answer from the options given below:

LIST-I	LIST-II
A. Point of local maxima is	I. -208
B. Point of local minima is	II. -83
C. Local maximum value is	III. 6
D. Local minimum value is	IV. 1

A-III, B-IV, C-I, D-II

A-IV, B-III, C-II, D-I

A-III, B-IV, C-II, D-I

A-III, B-II, C-I, D-IV

✅ Correct Option: 2

$f'(x) = 6x^2 - 42x + 36 = 6(x-1)(x-6)$ , so critical points are $x=1$ and $x=6$ .

Since $f''(x) = 12x - 42$ , $f''(1) < 0$ gives local maximum at $x=1$ (IV) and $f''(6) > 0$ gives local minimum at $x=6$ (III).

$f(1) = -83$ (II) and $f(6) = -208$ (I).

2025: 3 June Shift 1

Let f be a function defined by $f(x) = 2x^3 - 3x^2 - 36x + 2$ , then which of the following are correct?

(A) The critical points of f(x) are -2 and 3.

(B) The function f(x) increases in the interval $(3, \infty)$

(D) The function f(x) increases in the interval (-2,3)

Choose the correct answer from the options given below:

2023: 30 May Shift 3

The two curves $x^3 - 3xy^2 + 15 = 0$ and $3x^2 y - y^3 + 17 = 0$ :

2026: 22 May Shift 2

The cost function of a product is given by $C(x)=\frac{1}{3}x^3-45x^2-900x+36$ , where x is the number of units produced. In order to minimize the marginal cost, how many units of the product must be produced?

2022: 23 Aug Shift 1

Match list I with list II

List - I	List - II
A. Slope of tangent to the curve $y = x^3 - x$ at $x = 2$	I. $-2$
B. Slope of tangent to the curve $y = 3x^3 - 4x$ at $x = 0$	II. $11$
C. Slope of normal to the curve $y = \sin\theta$ at $\theta = \frac{\pi}{3}$	III. $2$
D. Slope of normal to the curve $y = \cos\theta$ at $\theta = \frac{\pi}{6}$	IV. $-4$

Choose the correct answer from the option given below :

2025: 15 May Shift 1

Which of the following are correct?

(A) The function $f(x) = 3x+12$ is increasing on R.

(B) The function $f(x) = e^{2x}$ is decreasing on R.

(D) The function $f(x) = x^3-3x^2+4x$ is increasing on R.

Choose the correct answer from the options given below:

2026: 19 May Shift 2

Let $f(x) = x^3 - 3ax$ , where $a > 0$ , then the values for which $f(x)$ is decreasing, are

Solution

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