I need help please !! f(x)=7x−2xln(x),x>0 . x-coordinate of absolute maximum = ?

Question

f(x)=7x−2xln(x),x>0 . x-coordinate of absolute maximum = ?

Answer 1

Let find derivative: f'(x) = 7 - 2ln(x) - 2x·1/x = 7 - l n(x) - 2 = 5 - ln(x). To get critical number set f'(x) = 0.

We get equation 5 - ln(x) = 0, ln(x) = 5. x = e⁵.

Because in domain of this function (x > 0) f''(x) = - 1/x and f''(e⁵) = - 1/e⁵ = - e^-5 < 0 then at x = e⁵ f(x) has absolute maximum.