Question

Describe the concavity of the function f(x) = 1/x^3
A. The graph is concave down for x < 0 and concave up for x > 0.
B. The graph is concave up for x < 0 and concave down for x > 0.
C. The graph is concave up for x < 0 and concave up for x > 0.
D. The graph is concave down for x < 0 and concave down for x > 0.

Answer 1

same steps as before

Answer 2

The second derivative describes the concavity of a function, so first lets take it.

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

f'(x) = (2x)(ln(x)) + (x^2)(1/x)
f'(x) = (2x)(ln(x)) + x

f''(x) = 2ln(x) + (2x)(1/x) + 1
f''(x) = 2ln(x) + 2 + 1
f''(x) = 2ln(x) + 3

Now lets plug in 1 for x into the function.

f''(1) = 2ln(1) + 3
f''(1) = 3

A concavity of POSITIVE 3 means that the graph is concave UP at x = 1. Hope that helps =)