Calculus

Application of Derivatives

Medium

core

If the minimum value of $a$ is $-\frac{k}{2}$ such that the function $f(x) = x^2 + ax + 5$ is increasing in [1, 2]. Then value of $k$ is

-4

-2

✅ Correct Option: 3

We have a function: $f(x) = x^2 + ax + 5$

We need to find the value of $k$ where the minimum value of $a$ is $-\frac{k}{2}$ , such that the function is increasing on the interval $[1, 2]$ .

A function is increasing when its derivative is non-negative throughout the interval.

For $f(x) = x^2 + ax + 5$ , the derivative is:

$f'(x) = 2x + a$

For the function to be increasing on $[1, 2]$ :

$f'(x) \geq 0$ for all $x \in [1, 2]$

$2x + a \geq 0$

$a \geq -2x$

The condition $a \geq -2x$ must hold for every value of $x$ in $[1, 2]$ .

Since $-2x$ is a decreasing function (it gets more negative as $x$ increases), we need to find where it's largest.

At $x = 1$ : $-2(1) = -2$

At $x = 2$ : $-2(2) = -4$

The largest value of $-2x$ on $[1, 2]$ is $-2$ (at $x = 1$ ).

Therefore: $a \geq -2$

The minimum value of $a$ is $-2$ .

The question states the minimum value of $a$ is $-\frac{k}{2}$ .

So:

$-\frac{k}{2} = -2$

$\frac{k}{2} = 2$

$k = 4$

Therefore, $k = 4$

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

Consider the function $f(x) = x^{\frac{1}{x}}$ . Its

2022: 25 May Shift 1

Match List I with List II

LIST I	LIST II
A. Maximum value of $f(x) = -\lvert x+1 \rvert + 3$	I. 6
B. Minimum value of $f(x) = (2x-1)^2 + 5$	II. 5
C. Maximum value of $f(x) = 6 - x^2$	III. no maximum value
D. Maximum value of $f(x) = x^3 + 1$	IV. 3

Choose the correct answer from the options given below:

2023: 30 May Shift 3

The rate of change in area of a triangle having sides 10 cm and 12 cm when the variable angle between them is $\theta = 60°$ , is :

2025: 13 May Shift 2

The function $f(x) = \frac{x}{2} + \frac{2}{x}, x \neq 0$ is increasing on

(A) $(-\infty, -2)$

(B) $(-2, 2)$

(D) $(-1, 1)$

Choose the correct answer from the options given below:

2026: 21 May Shift 2

The difference of absoulte maximum (largest) and absoulte minimum (smallest) values of the function $f(x) = 12x^{4/3} - 6x^{1/3}, x \in [-1, 1]$ is

If the minimum value of aaa is −k2-\frac{k}{2}−2k​ such that the function f(x)=x2+ax+5f(x) = x^2 + ax + 5f(x)=x2+ax+5 is increasing in [1, 2]. Then value of kkk is

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CUET Mathematics Past Year Questions (Topic-Wise)

If the minimum value of $a$ is $-\frac{k}{2}$ such that the function $f(x) = x^2 + ax + 5$ is increasing in [1, 2]. Then value of $k$ is