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

1 Answer
Oct 30, 2016

Answer:

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.