Find the minimum value(s) of the function f(x) = e^x - 2x

Question

what are the steps to solving this question?

Answer 1

To find minimum values, first take the derivative

e^x - 2x

e^x - 2

then find when the function is equal to 0

e^x = 2

x = ln2

This x value is a minimum if the derivative is negative when x is slightly less than 2, and the derivative is positive when x is slightly greater than 2.

plugging in ln1.99 to the derivative function, we find that the derivative is negative, meaning that x is decreasing

plugging in ln2.01 to the derivative function, we find that the derivative is positive, meaning that x is growing

this means that x = ln2 is a minimum

plug x = ln2 into the original equation

so:

e^ln2-2ln2 = y

Recall the identity that e^lnx = x

2 - 2ln2 = y

The minimum value of this function is 2 - 2ln2