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Question

Find the local maxima and local minima, if any of the following function. Also, find the local maximum and the local minimum values, as the case may be as follows.

h(x)=sinx+cosx,0<x<π2

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Solution

Given function is, h(x)=sinx+cosx,0<x<π2
h(x)=cosxsinx and h"(x)=sinxcosx
For maxima or minima put h'(x)=0
sinx=cosxsinxcosx=1tanx=1x=π4ϵ(0,π2)
At x=π4,h"(π4)=sinπ4cosπ4=sinπ4cosπ4=1212=22=2<0
Therefore, by second derivative test x=π4 is a point of local maxima and the local maximum value of h at x=π4 is
h(π4)=sinπ4+cosπ4=12+12=22=2


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