Consider the equation below.

Question

f(x) = x⁵ ln x

(a) What is the domain of the function?

(b) Find the interval(s) on which f is increasing. (Enter your answer using interval notation.)

Find the interval(s) on which f is decreasing. (Enter your answer using interval notation.)

(c) Find the x-coordinate(s) of any local minima. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

Local minima at x = 

Find the x-coordinate(s) of any local maxima. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

Local maxima at x = 

(d) Find the interval(s) on which f is concave up. (Enter your answer using interval notation.)

Find the interval(s) on which f is concave down. (Enter your answer using interval notation.)

Find the x-coordinate(s) of any inflection points. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

Inflection point(s) at x =

Answer 1

(a) Domain: (0, ∞)

(b) f'(x) = 5x⁴lnx + x⁴ > 0; x⁴(1 + 5lnx)> 0; 1 + 5lnx > 0; lnx > - 1/5; x > e^-1/5; (e^-1/5, ∞) f(x) increasing

So, f'(x) < 0, 1 + 5lnx < 0; 0< x < e^-1/5 Interval (0, e^-1/5) f(x) decreasing

(c) f'(x) = 0; 1 + 5lnx = 0; x = e^-1/5. At x = e^-1/5 f'(x) changed sign from - to +

So, at x = e^-1/5 f(x) has local min and local max DNE

(d) f''(x0 = 20x³lnx + 9x³ = x³(20lnx + 9) = 0; x = e^-9/20

f''(x) > 0 for x > e^-9/20; so (e^-9/20, ∞) f(x) concave up; (0, e^-9/20) f''(x) < 0 and f(x) concave down

At x = e^-9/20 _{inflection point}