0Solution:
There are different ways to solve this problem. I will discuss here the Derivative method.
Steps
1. Find the first derivative of the given function.
2. Equate the result of the first derivative to zero to get the critical points.
3. Find the second derivative of the function and check it is positive or negative or zero at the critical points.
Now Solving:
Let the given function is f(x) that is f(x) = sinx +cosx
Finding the first derivative
f'(x) = cosx - sinx
Equating it to zero
f'(x) =0
or, cosx - sinx =0
or, tanx =1
or, x = 45 Degree
Finding the Second Derivative
f"(x) = -(sinx+cox)
At x= 45 , f"(x) <0
That means x = 45 is the point of maxima.
Hence the maximum value of the function f(x) is
f(x= 45)
= sin 45 + cos 45
= Square Root 2 (Ans)
