The function f(x)= (8x-7)e^2x has one critical number. Find it

Answer 1

A critical point in the domain of x is a value of x at which the function is not differentiable or where the derivative is zero.
See: http://en.wikipedia.org/wiki/Critical_point_%28mathematics%29

For the case in point:
f(x)= (8x-7)e^2x
f'(x) can be found using the product rule and the chain rule as
f'(x):=2(8x-3)*e^(2x)
Setting f'(x)=0 and solving for x, we get e^2x=0 or x=3/8.
Since the range of e^2x excludes 0, it is a solution to be rejected. So the unique point required is x=3/8.