GATE (TF) Textile 2019 Question Paper Solution | GATE/2019/TF/02

Question 02 (Textile Technology & Fibre Science)

For x in [0, \pi], is maximum value of (\sin x+ \cos x) is

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F=(\sin x+ \cos x)

Differentiate it both side with respect to x

\frac{dF}{dx}=(\cos x-\sin x)

For maxima minima , \frac{dF}{dx}=0

(\cos x-\sin x)=0

cos x=sin x


Again differentiate-

\frac{d^2F}{dx^2}=(\-sin x-\cos x)

\frac{d^2F}{dx^2}=(\-sin\frac{\pi}{4} -\cos \frac{\pi}{4})=(-\frac{1}{\sqrt 2}-\frac{1}{\sqrt 2})=-\sqrt 2

Hence function is minima at x=\frac{\pi}{4}

F=(\sin x+ \cos x)

By putting x=\frac{\pi}{4}

F=(\sin \frac{\pi}{4}+ \cos \frac{\pi}{4})

F=(\frac{1}{\sqrt 2}+\frac{1}{\sqrt 2})

F=\frac{2}{\sqrt 2}

F=\sqrt 2 (Ans)

This is minimum value of the function.

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