The function $f(x) = \tan x - 4x$ is strictly decreasing on
- $\left( - \frac{\pi}{3} , \frac{\pi}{3} \right)$
- $\left( \frac{\pi}{3} , \frac{\pi}{2} \right)$
- $\left( - \frac{\pi}{3} , \frac{\pi}{2} \right)$
- $\left( \frac{\pi}{2} ,\pi \right)$
The Correct Option is A
Solution and Explanation
\(f(x)=\tan x-4 x \Rightarrow f'(x)=\sec ^{2} x-4\)
When \(\frac{-\pi}{3} < x < \frac{\pi}{3}, 1 < \sec x < 2\)
Therefore, \(1 < \sec ^{2} x < 4\)
\(\Rightarrow-3 < \left(\sec ^{2} x-4\right) < 0\)
Thus, for \(\frac{-\pi}{3} < x < \frac{\pi}{3}, f'(x) < 0\)
Hence, \(f\) is strictly decreasing on
\(\left(\frac{-\pi}{3}, \frac{\pi}{3}\right)\)
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- Solutions
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