Find the local maxima or local minima, if any, of the function `f(x)=sinx+cosx ,\ \ 0 lt x lt pi/2` using the first derivative test.

Explanation

`h(x)=sin X+cos X_{1} 0<{X}`
therefore h^{prime}(x)=cos x-sin x `
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h^{prime}(x)=0 Rightarrow sin x=cos x Rightarrow tan x=1 Rightarrow x=frac{pi}{4} in(0, frac{pi}{2}) `
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h^{prime}(x)=-sin x-cos x=-(sin x+cos x) `
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h^{*}(frac{pi}{4})=-(frac{1}{sqrt{2}}+frac{1}{sqrt{2}})=-frac{2}{sqrt{2}}=-sqrt{2}<0`


Therefore, by second derivative test, `x=frac{pi}{4}` is a point of local maxima and the local maximum value of h at `x=frac{pi}{4} is h(frac{pi}{4})=sin frac{pi}{4}+cos frac{pi}{4}=frac{1}{sqrt{2}}+frac{1}{sqrt{2}}=sqrt{2}.`
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