Find the maximum value of the function `f(x) =sin x + cos x`.

Explanation

`f(x) = sin x + cos x`
` rArr f'(x) = cos x- sin x`
For maxima or minima
` f'(x) = 0`
` rArr cos x - sn x= 0`
` rArr cos x = sin x`
` rArr x = pi/4`
Now ` f''(x) =-sin x - cos x `
`"At " x = pi//4`,
` f''(pi/4) =- sin pi/4 - cos pi/4`
` = - 1/sqrt2= 1/sqrt2 =- sqrt2 lt 0 `
` :. f(x) " is maximum at " x pi//4`
and maximum value = ` f(pi/4)`
`=sin pi/4 + cos pi/4`
` = 1/sqrt2 + 1/sqrt2 = sqrt2`.