Calculus

Application of Derivatives

Easy

core

The function $f(x) = x^2 - x + 1$ is

Increasing on $(\frac{-1}{2}, 1)$ and decreasing on $(0, \frac{1}{2})$

Increasing on $(\frac{1}{2}, \infty)$ and decreasing on $(-\infty, \frac{1}{2})$

Increasing on $(-\infty, \frac{1}{2})$ and decreasing on $(\frac{1}{2}, \infty)$

Increasing on $(-\infty, 1)$ and decreasing on $(1, \infty)$

✅ Correct Option: 2

Given the function $f(x) = x^2 - x + 1$

To find where the function is increasing or decreasing, find the derivative:

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

Setting $f'(x) = 0$ to find the critical point:

$2x - 1 = 0$

$2x = 1$

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

For $x < \frac{1}{2}$ , test $x = 0$ :

$f'(0) = 2(0) - 1 = -1$

Since $f'(0) < 0$ , the function is decreasing on $(-\infty, \frac{1}{2})$

For $x > \frac{1}{2}$ , test $x = 1$ :

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

Since $f'(1) > 0$ , the function is increasing on $(\frac{1}{2}, \infty)$

The function is:

Decreasing on $(-\infty, \frac{1}{2})$

Increasing on $(\frac{1}{2}, \infty)$

2023: 22 May Shift 3

If the rate of change of area of a circle is equal to the rate of change of its diameter, then its radius is equal to :

2026: 12 May Shift 2

The total revenue received from the sale of $x$ units of a product is given by $R(x)=24x+5x^2+3$ . The marginal revenue when $x=10$ is,

2026: 22 May Shift 2

If the local maximum value of the function $f(x)=x^3-2\lambda x^2+\lambda^2 x, \lambda>0$ is 108, Then the local minimum value of the function occurs at

2026: 22 May Shift 2

If the function $f(x)$ defined by $f(x) = x^3 - 3x + 100, (x \in \mathbb{R})$ , (Where $\mathbb{R}$ is set of real numbers), then $f(x)$ has

2025: 16 May Shift 1

If the interval in which the function f(x) = $\frac{x}{x^2+1}$ is strictly increasing is (-a, a), then a is equal to

2025: 16 May Shift 1

Match List-I with List-II

Consider the function f(x) = 2x³ - 21x² + 36x + 80, x∈[0, 6]. Then

List-I	List-II
(A) one of its critical points is at x =	(I) -28
(B) Its absolute maximum value is	(II) -42
(C) Its absolute minimum value is	(III) 97
(D) Its second derivative at x = 0 is	(IV) 6

Choose the correct answer from the options given below:

The function f(x)=x2−x+1f(x) = x^2 - x + 1f(x)=x2−x+1 is

Solution

Related questions:

Keyboard Shortcuts

CUET Mathematics Past Year Questions (Topic-Wise)

The function $f(x) = x^2 - x + 1$ is