If f''(x) > 0 on (a,b), the function is concave up on that interval. Which choice best expresses this?

If f''(x) > 0 on (a,b), the function is concave up on that interval. Which choice best expresses this?

• A. concave down
• B. concave up
• C. linear
• D. has a maximum everywhere

Answer: (B)

Concavity is determined by the second derivative: when f''(x) > 0, the graph is concave up on that interval. This means the slope of the tangent line is increasing as x increases, so the curve bends upward like a cup. An example you can picture is f(x) = x^2, where f''(x) = 2 > 0 everywhere, giving a clear concave-up shape. So the best description here is concave up because the positive second derivative indicates the graph bends upward and the tangent slopes are getting steeper as x grows. A concave-down graph would have f''(x) < 0, a linear function would have f''(x) = 0, and a function with a maximum everywhere cannot have f'' > 0 on the interval.

Concavity is determined by the second derivative: when f''(x) > 0, the graph is concave up on that interval. This means the slope of the tangent line is increasing as x increases, so the curve bends upward like a cup. An example you can picture is f(x) = x^2, where f''(x) = 2 > 0 everywhere, giving a clear concave-up shape.

So the best description here is concave up because the positive second derivative indicates the graph bends upward and the tangent slopes are getting steeper as x grows. A concave-down graph would have f''(x) < 0, a linear function would have f''(x) = 0, and a function with a maximum everywhere cannot have f'' > 0 on the interval.