Question

How do you find the maximum value of `f(x) = sinx+cosx`?

Answer 1

Please see the explanation.

Explanation:

The x coordinates of extrema can be found by, computing the first derivative, setting that equal to zero, and then solving for x:

Compute the first derivative:

`f'(x) = cos(x) - sin(x)`

Set equal to zero:

`0 = cos(x) - sin(x)`

Solve for x:

`cos(x) = sin(x)`

`1 = sin(x)/cos(x)`

`1 = tan(x)`

`x = tan^-1(1)`

`x = pi/4`

Because the tangent function has a period of `pi`, the value, 1, repeats at every integer multiple of `pi`:

`x = pi/4 + npi` where `n = ...,-3,-2, -1, 0, 1, 2,3,...`

To determine whether this is a maximum, perform the second derivative test, using one of the values:

`f''(x) = -cos(x) - sin(x)`

Evaluate at `pi/4`

`f''(pi/4) = -cos(pi/4) - sin(pi/4) = -sqrt(2)`

The value is a negative, therefore, we have found a maximum.