Question

Please help!!! this is a multiple choice. determine the minimum value of the function f(x)=e^(-x)-2e^x on the interval -1≤x≤2.?

the choices are
a) 1.98
b) 7.12
c) -14.64
d) -5.07

Answer 1

The answer is the minimum on the interval is `f(2)=e^{-2}-2e^2` which isn't really a choice, but (c) is a good approximation.

Explanation:

`f(x) = e^{-x} - 2e^x`

`f'(x) = - e^{-x} - 2 e^ x`

That derivative is clearly negative everywhere so the function is decreasing over the interval. So its minimum value is `f(2)=e^{-2}-2e^2`. If I was a stickler (which I am) I'd answer None of the Above because there's no way that transcendental quantity can equal one of those rational values. But we succumb to approximation culture and get out the calculator, which says

`f(2) approx -14.6428` which is choice (c)