The sum of the absolute maximum and absolute minimum values of the function \(\begin{array}{l} f\left(x\right)=\tan^{-1}\left(\sin x-\cos x\right) \end{array}\)in the interval\( [0, π]\) is
- 0
- \(tan^{-1}\frac{1}{\sqrt{2}}-\frac{π}{4}\)
- \(cos^{-1}\frac{1}{\sqrt{3}}-\frac{π}{4}\)
- \(-\frac{π}{12}\)
The Correct Option is C
Solution and Explanation
\(f(x)=tan^{−1}(sinx−cosx)\)
Let \(g(x)=sinx−cosx\)
=\(\sqrt{2}sin(x−\frac{π}{4})\) and \(x−\frac{π}{4}∈[−\frac{π}{4},\frac{3π}{4}]\)
∴ g\((x)∈[−1,\sqrt2]\)
and \(tan^{−1}x\) is an increasing function
∴ \(f(x)∈[tan^{−1}(−1),tan^{−1}\sqrt2] ∈[−\frac{π}{4},tan^{−1}\sqrt2]\)
∴ Sum of \(f_{max}\) and \(f_{min}\)=\(tan^{−1}\sqrt{2}−\frac{π}{4}\)
= \(cos^{−1}(\frac{1}{\sqrt{3}})−\frac{π}{4}\)
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Concepts Used:
Maxima and Minima
What are Maxima and Minima of a Function?
The extrema of a function are very well known as Maxima and minima. Maxima is the maximum and minima is the minimum value of a function within the given set of ranges.
There are two types of maxima and minima that exist in a function, such as:
- Local Maxima and Minima
- Absolute or Global Maxima and Minima