If f(x)=3\(\sqrt[3]{x^2}-x^2\), then
- f has no extrema
- f is maximum at two points x=1 and x=-1
- f is minimum at x=0
- f has maximum at x=1 only
The Correct Option is B, C
Solution and Explanation
Given: We are analyzing the function to determine where it attains maximum and minimum values.
From the graph or the nature of the function, it is observed that:
- Maximum value occurs at: $x = 1$ and $x = -1$
- Minimum value occurs at: $x = 0$
These are the critical points of the function — where the first derivative is zero or undefined, and the second derivative test or graph shape confirms the nature of the extrema.
Correct option(s):
(B): $f$ is maximum at two points $x = 1$ and $x = -1$ ✅
(C): $f$ is minimum at $x = 0$ ✅
Top Questions on Functions
- If $ f(x) = 2x^2 - 3x + 5 $, find $ f(3) $.
- Determine those values of $x$ for which $f(x) = \frac{2}{x} - 5$, $x \ne 0$ is increasing or decreasing.
Let A be the set of 30 students of class XII in a school. Let f : A -> N, N is a set of natural numbers such that function f(x) = Roll Number of student x.
Give reasons to support your answer to (i).- Assertion (A): Let \( A = \{ x \in \mathbb{R} : -1 \leq x \leq 1 \} \). If \( f : A \to A \) be defined as \( f(x) = x^2 \), then \( f \) is not an onto function.
Reason (R): If \( y = -1 \in A \), then \( x = \pm \sqrt{-1} \notin A \). Find the domain of the function \( f(x) = \cos^{-1}(x^2 - 4) \).
Questions Asked in WBJEE exam
- Which logic gate is represented by the following combination of logic gates?
- WBJEE - 2025
- Logic gates
- If \( {}^9P_3 + 5 \cdot {}^9P_4 = {}^{10}P_r \), then the value of \( 'r' \) is:
- WBJEE - 2025
- permutations and combinations
- If \( 0 \leq a, b \leq 3 \) and the equation \( x^2 + 4 + 3\cos(ax + b) = 2x \) has real solutions, then the value of \( (a + b) \) is:
- WBJEE - 2025
- Trigonometric Equations
- Let \( f(\theta) = \begin{vmatrix} 1 & \cos \theta & -1 \\ -\sin \theta & 1 & -\cos \theta \\ -1 & \sin \theta & 1 \end{vmatrix} \). Suppose \( A \) and \( B \) are respectively the maximum and minimum values of \( f(\theta) \). Then \( (A, B) \) is equal to:
- If \( a, b, c \) are in A.P. and if the equations \( (b - c)x^2 + (c - a)x + (a - b) = 0 \) and \( 2(c + a)x^2 + (b + c)x = 0 \) have a common root, then
- WBJEE - 2025
- Quadratic Equations
Concepts Used:
Functions
A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A & B be any two non-empty sets, mapping from A to B will be a function only when every element in set A has one end only one image in set B.
Kinds of Functions
The different types of functions are -
One to One Function: When elements of set A have a separate component of set B, we can determine that it is a one-to-one function. Besides, you can also call it injective.
Many to One Function: As the name suggests, here more than two elements in set A are mapped with one element in set B.
Moreover, if it happens that all the elements in set B have pre-images in set A, it is called an onto function or surjective function.
Also, if a function is both one-to-one and onto function, it is known as a bijective. This means, that all the elements of A are mapped with separate elements in B, and A holds a pre-image of elements of B.
Read More: Relations and Functions