Answer to Question #103926 in Calculus for Martin

Question

Find the exact global maximum and minimum values of the function g(x)=7x−x2−15 if its domain is all real numbers.

Answer 1

$f(x)=7x-x^2-15$

To find stationary points, the function is differentiated and equated to zero.

$f'(x)=7-2x$

$7-2x=0$

$7=2x, x=3.5$

There is one stationary point.

$7\times{3.5}-3.5^2-15=-2.75$

The critical point is (3.5,-2.75).

To determine whether it is a maximum or minimum, the second derivative is required.

$f''=-2$

Since the second derivative is negative, the stationary point is a maximum. Since there are no other maximum stationary points, (3.5,-2.75) is a global maximum. The function has no minimum point.