Calculus

Application of Derivatives

Medium

The given function $f(x) = x^5 - 5x^4 + 5x^3 - 1$ ; has/have

(a) local maxima at $x = 1$

(b) local maximum value is 0

(c) local minimum at $x = 3$

(d) local minimum value is $-28$

(e) The point of inflexion is $x = 1$

Choose the correct answer from the options given below

(a), (b) only

(a), (b), (c) only

(a), (b), (c), (d) only

(a), (c), (e) only

✅ Correct Option: 3

$f'(x) = 5x^4 - 20x^3 + 15x^2 = 5x^2(x-1)(x-3)$ . Critical points: $x=0,1,3$ .

$f''(x) = 10x(2x^2-6x+3)$ . $f''(0)=0$ (inflexion), $f''(1)<0$ (max), $f''(3)>0$ (min).

$f(1) = 1-5+5-1 = 0$ (local max value).

$f(3) = 243-405+135-1 = -28$ (local min value).

So (a), (b), (c), (d) are correct; (e) is wrong since inflexion is at $x=0$ .

2025: 15 May Shift 1

If $f(x) = a \log_e|x| + bx^2 + x$ has critical points at $x = -2$ and $x = 1$ , then

2026: 18 May Shift 2

Match the LIST-I with LIST-II

LIST-I Function $f(x)$	LIST-II $f(x)$ is increasing in the interval
A. $f(x)=-x^2-2x+3$	I. $(-\infty,2)$
B. $f(x)=x^2+4x+1$	II. $(1,\infty)$
C. $f(x)=2x^2-4x+3$	III. $(-\infty,-1)$
D. $f(x)=-3x^2+12x+1$	IV. $(-2,\infty)$

Choose the correct answer from the options given below:

2022: 25 May Shift 1

The approximate volume of a cube of side a meters on increasing the side by 4% is:

2025: 27 May Shift 1

The interval on which the function $f(x) = x^3 + 2x^2 - 1$ is decreasing, is

2025: 21 May Shift 2

Match List-I with List-II

Choose the correct answer from the options given below:

2026: 12 May Shift 2

A function $f(x)$ is defined as:

$f(x) = x^3-6x^2+9x+15$

For the function $f(x)$ , which of the following statements are correct ?

(A) The critical points are $x=1$ and $x=3$ .

(B) $x=1$ is a point of local minimum.

(D) $x=3$ is a point of local minimum.

Choose the correct answer from the options given below:

Solution