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
Given function is, h(x)=sinx+cosx,0<x<π2
∴h′(x)=cosx−sinx and h"(x)=−sinx−cosx
For maxima or minima put h'(x)=0
⇒sinx=cosx⇒sinxcosx=1⇒tanx=1⇒x=π4ϵ(0,π2)
At x=π4,h"(π4)=−sinπ4−cosπ4=−sinπ4−cosπ4=−1√2−1√2=−2√2=−√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=1√2+1√2=2√2=√2