For the function f(x)=ax^3-9ax^2+9x+3, x R, then which of the following statements are True? (A) f'(x)=3[ax^2-6ax+3] (B) It is increasing for a[0,1/3]\ x R (C) It is increasing for a(-∞,0)\ x R (D) It is increasing for a R Choose the correct answer from the options given below: Option 1: (A) and (C) only Option 2: (A) and (B) only Option 3: (B) and (C) only Option 4: (C) and (D) only

Answer: (A) and (B) only. Solution: f'(x)=3ax^2-18ax+9=3(ax^2-6ax+3), so (A) is true. For f'(x)≥0 for all real x we need a≥0 and discriminant 36a^2-12a≤0, i.e. 0≤ a≤13. Hence (B) holds while (C) and (D) fail.

Calculus

Application of Derivatives

Hard

core

For the function $f(x)=ax^3-9ax^2+9x+3, x\in R$ , then which of the following statements are True?

(A) $f'(x)=3[ax^2-6ax+3]$

(B) It is increasing for $a\in\left[0,\dfrac{1}{3}\right]\ \forall x\in R$

(C) It is increasing for $a\in(-\infty,0)\ \forall x\in R$

(D) It is increasing for $\forall a\in R$

Choose the correct answer from the options given below:

(A) and (C) only

(A) and (B) only

(B) and (C) only

✅ Correct Option: 2

$f'(x)=3ax^2-18ax+9=3(ax^2-6ax+3)$ , so (A) is true. For $f'(x)\ge0$ for all real $x$ we need $a\ge0$ and discriminant $36a^2-12a\le0$ , i.e. $0\le a\le\dfrac13$ . Hence (B) holds while (C) and (D) fail.

2026: 29 May Shift 2

The interval in which the function $f(x)=x^3-12x^2+36x+20$ is increasing, is

2026: 25 May Shift 1

Which of the following statements are correct ?

A. The function $f(x) = a^x$ is increasing on $(-\infty,\infty)$ if $a>1$ .

B. $f(x) = x^2 - 2x$ is decreasing in $(-\infty,1)$ .

C. $f(x) = x(x-3)^2$ increases for $1 \le x \le 3$ .

D. $f(x) = e^{-x}$ decreases in $[0,\infty)$ .

Choose the correct answer from the options given below:

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

2025: 16 May Shift 1

For x ∈ ℝ - {0}, the function f(x) = $\frac{3}{x}$ + 7 is decreasing when

2025: 14 May Shift 2

The greatest possible value of ' $a$ ' such that the function $f(x) = x^2 + a x + 1$ is always decreasing in the interval [1, 2] is:

2026: 18 May Shift 2

The total revenue received from the sale of $x$ units of a product is given by $R(x)=542+\dfrac{x^2+3}{4}$ . What will be the marginal revenue when $x=24$ ?

Solution

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