What is the maximum value of the function sin x + cos x ?
The given function is,
f ( x ) = sin x + cos x
Differentiate the function with respect to x,
f ′ ( x ) = cos x − sin x (1)
Put f ′ ( x ) = 0 ,
cos x − sin x = 0 cos x = sin x tan x = 1 x = π 4 , 5 π 4 , ⋯
Differentiate equation (1) with respect to x,
f ″ ( x ) = − sin x − cos x = − ( sin x + cos x )
The value of f ″ ( x ) is negative only when the value of f ( x ) is positive which is only possible when the function lies in first quadrant in the interval ( 0 , π 2 ) . For this interval, the value of x is π 4 .
The maximum value of the given function is,
f ( π 4 ) = sin π 4 + cos π 4 = 1 2 + 1 2 = 2 2 = 2
Therefore, the maximum value of the given function is 2 .
The given function is,
Differentiate the function with respect to x,
Put
Differentiate equation (1) with respect to x,
The value of
The maximum value of the given function is,
Therefore, the maximum value of the given function is
