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 $x = 0$ ✅
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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