Fill in the blanks: If f(x)=| - |, then f' ( 3 )= _______. - Vidyadip

Question

Fill in the blanks:
If $\text{f(x)}=|\cos\text{x}-\sin\text{x}|,$ then $\text{f}'\Big(\frac{\pi}{3}\Big)=$ _______.

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Answer

If $\text{f(x)}=|\cos\text{x}-\sin\text{x}|,$ then $\text{f}'\Big(\frac{\pi}{3}\Big)=\frac{\sqrt{3}+1}{2}.$
Solution:
We have, $\text{f(x)}=|\cos\text{x}-\sin\text{x}|$
For $\frac{\pi}{4}<\text{x}<\frac{\pi}{2},\cos\text{x}<\sin\text{x}$
$\therefore\ \text{f(x)}=\sin\text{x}-\cos\text{x}\text{ for }\frac{\pi}{4}<\text{x}<\frac{\pi}{2}$
$\Rightarrow\ \text{f}'(\text{x})=\cos\text{x}+\sin\text{x}$
$\Rightarrow\ \text{f}'\Big(\frac{\pi}{3}\Big)=\frac{\sqrt{3}+1}{2}$

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