What is the maximum value of sinx + cosx ?

Solution:

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)