Help with business calc!

Question

Suppose g'(x)= x^4 +x^2 +1. Find the x values in the interval [-3,4] where g(x) takes its minimum. 

 

The answer is at x=-3 but why? I'm not sure how to get this.

Answer 1

The values in the interval where g is at a minimum are either at the endpoints of the interval, or if there's a critical (minimum) value in the interval.

 

To try to find a critical value, set it equal to 0, and solve. In this case, there is no (real) solution. There is no critical value, so the minimum must be an endpoint.

 

If you look at the value g' can take, you'll see that it is always > 0 (even powers + 1.) Therefore the original function g is always increasing, and it's minimum value is on the left endpoint.