Answer the questions

Question

f(x) =6x^2ln(x)

a) List all the critical values of f(x)

b ) use the intevral notation to indicate where f(x) is increasing

f) use the interval notation to indicate where f(x) is concave up

g) use interval notation to indicate where f(x) is concave down

h) list the values x of all inflection points of f.

Answer 1

f(x) = 6x²ln(x)

(a)

f’(x)= 12xln(x) + 6x²/x = 12xln(x) + 6x = 6x(2ln(x) +1)

f’(x)= 0 when 2ln(x) + 1 = 0 which means

f(e^–1/2) = –3/e

f(x) and f’(x) are both undefined at x ≤ 0

(b)

(f), (g), and (h)

f’’(x) = 12ln(x) +12x/x +6 = 12ln(x)+18 = 6(2ln(x) + 3)

f(x) has an inflection point at x = e^–3/2

On the interval (0, e^–3/2) f''(x) < 0 so f(x) is concave down

On the interval (e^–3/2,∞) f''(x) > 0 so f(x) is concave up