Calculus

Application of Derivatives

Medium

core

For $x \in \mathbb{R}$ , if $f(x) = -(x-1)^2 + 2$ , then

(A) $f$ is an increasing function on $(-\infty, 1]$

(B) $f$ has no critical points

(C) $f$ has a maximum value at $x = 1$

(D) $f$ has a minimum value at $x = 1$

Choose the correct answer from the options given below:

(B) and (D) only

(B), (C) and (D) only

(A) and (C) only

(A) and (B) only

✅ Correct Option: 3

The function $f(x) = -(x-1)^2 + 2$ is a quadratic function in vertex form $f(x) = a(x-h)^2 + k$ .

From the given function:

Vertex: $(h, k) = (1, 2)$
Coefficient: $a = -1 < 0$

Since $a < 0$ , the parabola opens downward.

For a downward-opening parabola with vertex at $x = 1$ :

The function increases on $(-\infty, 1]$
The function decreases on $[1, \infty)$

Statement (A) is TRUE.

To find critical points, take the derivative:

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

Setting $f'(x) = 0$ :

$-2(x-1) = 0$

$x = 1$

A critical point exists at $x = 1$ .

Statement (B) is FALSE.

Since the parabola opens downward, the vertex represents the maximum point.

At $x = 1$ :

$f(1) = -(1-1)^2 + 2$

$f(1) = 2$

The function has a maximum value of $2$ at $x = 1$ .

Statement (C) is TRUE.

Since the parabola opens downward, the vertex represents a maximum, not a minimum.

As $x \to \pm\infty$ , $f(x) \to -\infty$ , so no minimum value exists.

Statement (D) is FALSE.

The correct statements are (A) and (C) only.

2025: 21 May Shift 2

For the function, $f(x) = \frac{-3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 - 350$ , which of the following statements are correct?

(A) $x = -3$ and $x = -5$ are the only critical points of the given function.

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

(D) $x = -5$ is a point of local maximum.

Choose the correct answer from the options given below:

2026: 19 May Shift 1

The maximum area of a rectangle inscribed in a circle of radius $3\sqrt{2}$ cm, is:

2026: 19 May Shift 1

The absolute maximum value of the function $f(x) = 2x^3 - 15x^2 + 36x + 7$ on the interval [1,5] is

2022: 7 Aug Shift 2

If $2x + y = 6$ then the maximum value of $x^2 y$ is :

2026: 19 May Shift 1

For the function $f(x) = \frac{x^4}{4} - 2x^3 + \frac{11}{2}x^2 - 6x$ , which of the following statements are TRUE?

(A) Critical points of $f(x)$ are $x = 1, 2$ and $3$ .

(B) $f(x)$ is increasing over $(1,2)$ .

(D) $f(x)$ is increasing over $(3,\infty)$ .

Choose the correct answer from the options given below:

2025: 19 May Shift 1

The minimum value of the function $f(x) = x^3 + (10-x)^3$ occurs at:

Solution

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