Calculus

Application of Derivatives

Medium

common

The function $f(x) = x + \frac{1}{x}$ has

local maxima at $x = -1$

local minima at $x = -1$

local maxima at $x = 1$

neither local maxima nor local minima

✅ Correct Option: 1

The function $f(x) = x + \frac{1}{x}$ has a local maximum or minimum where the derivative equals zero.

$f'(x) = 1 - \frac{1}{x^2}$

Setting the derivative equal to zero:

$1 - \frac{1}{x^2} = 0$

$1 = \frac{1}{x^2}$

$x^2 = 1$

$x = 1$ or $x = -1$

To determine whether each critical point is a maximum or minimum, use the second derivative test.

$f''(x) = \frac{2}{x^3}$

When $f''(x) < 0$ , the function has a local maximum.

When $f''(x) > 0$ , the function has a local minimum.

At $x = -1$ :

$f''(-1) = \frac{2}{(-1)^3}$

$f''(-1) = \frac{2}{-1}$

$f''(-1) = -2$

Since $f''(-1) < 0$ , the function has a local maximum at $x = -1$ .

At $x = 1$ :

$f''(1) = \frac{2}{(1)^3}$

$f''(1) = 2$

Since $f''(1) > 0$ , the function has a local minimum at $x = 1$ .

Therefore, the function has a local maximum at $x = -1$ and a local minimum at $x = 1$ .

2026: 18 May Shift 2

For the function $f(x)=x^3-2x^2-4x-5$ , which of the following statement is not correct ?

2025: 27 May Shift 1

The largest open interval in which the function $f(x) = 4x^3 - 5x^2 - 8x + 12$ increases, is:

2026: 25 May Shift 1

The points in the interval $[0,2\pi]$ , where the function $f(x) = \cos2x + 2\sin x\cos x$ , attains its local minima, is/are:

A. $\pi/8$

B. $5\pi/8$

C. $13\pi/8$

D. $11\pi/8$

Choose the correct answer from the options given below:

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: 23 May Shift 3

The critical points of $f(x) = x^3 + x^2 + x + 1$ are

2025: 30 May Shift 2

The function $f: R \rightarrow R$ (where $R$ is set of real numbers) defined as

$f(x) = x^2 + 2x$ is

The function f(x)=x+1xf(x) = x + \frac{1}{x}f(x)=x+x1​ has

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The function $f(x) = x + \frac{1}{x}$ has