If a function is concave up on an interval, what does that say about its second derivative on that interval?

If a function is concave up on an interval, what does that say about its second derivative on that interval?

• A. f''(x) > 0
• B. f''(x) < 0
• C. f''(x) = 0
• D. f''(x) undefined

Answer: (A)

Concave up means the graph bends upward, so the slopes of the tangent lines are increasing as x increases. The second derivative tells you how fast those slopes are changing. If the function is twice differentiable on that interval, concave up forces the second derivative to be positive there, i.e., f''(x) > 0. A quick check: f(x) = x^2 has f''(x) = 2, which is positive everywhere, and its graph is concave up on all of R. Of course, if the second derivative doesn’t exist at some point, you can’t assign a value there, but where it exists, concave up implies a positive second derivative.

Concave up means the graph bends upward, so the slopes of the tangent lines are increasing as x increases. The second derivative tells you how fast those slopes are changing. If the function is twice differentiable on that interval, concave up forces the second derivative to be positive there, i.e., f''(x) > 0.

A quick check: f(x) = x^2 has f''(x) = 2, which is positive everywhere, and its graph is concave up on all of R. Of course, if the second derivative doesn’t exist at some point, you can’t assign a value there, but where it exists, concave up implies a positive second derivative.