The function f(x)=x^2-x√2 is strictly decreasing in the interval : Option 1: (-∞,√2) Option 2: [√2,∞) Option 3: (-∞,1/√2] Option 4: (-∞,1]

Answer: (-∞,1/√2]. Solution: f'(x)=2x-2, which is negative when x<(2)/2=1/(2). So the function decreases on (-∞,1/(2)], the vertex being the turning point.

Calculus

Application of Derivatives

Easy

applied

The function $f(x)=x^2-x\sqrt{2}$ is strictly decreasing in the interval :

$(-\infty,\sqrt{2})$

$[\sqrt{2},\infty)$

$\left(-\infty,\frac{1}{\sqrt{2}}\right]$

$(-\infty,1]$

✅ Correct Option: 3

$f'(x)=2x-\sqrt2$ , which is negative when $x<\frac{\sqrt2}{2}=\frac{1}{\sqrt2}$ .

So the function decreases on $\left(-\infty,\frac{1}{\sqrt2}\right]$ , the vertex being the turning point.

2026: 31 May Shift 2

Let $f(x)$ be a function defined on an interval $[a,b]$ such that $f(x)$ is differentiable on $(a,b)$ . Then

Match the LIST-I with LIST-II

	LIST-I Where $I$ is subset of $[a,b]$		LIST-II
A.	function $f(x)$ is increasing on interval $I$	I.	$f'(x) < 0\ \ \forall x \in I$
B.	function $f(x)$ has local maxima at $x = a$	II.	$f'(a) = 0\ \&\ f''(a) > 0$
C.	function $f(x)$ has local minima at $x = a$	III.	$f'(x) > 0\ \ \forall x \in I$
D.	function $f(x)$ is decreasing on interval $I$	IV.	$f'(a) = 0\ \&\ f''(a) < 0$

Choose the correct answer from the options given below:

2025: 14 May Shift 1

If the maximum value of the function $f(x) = \frac{\log_ex}{x}$ , $x > 0$ occurs at $x = a$ , then $a^2f''(a)$ is equal to

2025: 15 May Shift 1

If the function $f(x) = 2x^3 + 9x^2 + 12x-1$ is given,then $f(x)$ have

2025: 15 May Shift 2

The interval on which the function $f(x) = x^4 - \frac{x^3}{3}$ is strictly decreasing, is:

2026: 19 May Shift 2

If the sum of the length of the hypotenues and a side of a right angled triangle is L cm then the angle between hypotenuse and other side of the triangle for which the area of the triangle is maximum, is:

2026: 19 May Shift 1

For $f(x) = x^3 - 6x^2 + 9x + 15$ , which of the following statements are TRUE?

(A) $f'(x) = 3x^2 - 12x + 15$ .

(B) The critical points are 3 and 1.

(D) Local minimum value is 19.

Choose the correct answer from the options given below:

The function f(x)=x2−x2f(x)=x^2-x\sqrt{2}f(x)=x2−x2​ is strictly decreasing in the interval :

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The function $f(x)=x^2-x\sqrt{2}$ is strictly decreasing in the interval :