Calculus - Curve Sketching

Question

A graph of the curve y = f(x) is shown below.

A point on the graph, where f '(x) < 0 and f ''( x) < 0 is

1. B
2. D
3. A
4. C

Answer 1

1. The first derivative tells us about the slope of the graph.
2. If f'(x) is positive, the graph has a positive slope and is therefore increasing at x.
3. If f'(x) is negative, the graph has a negative slope and is therefore decreasing at x.
4. The second derivative tells us about the concavity of the graph.
5. If f''(x) is positive, the graph is concave up.
6. If f''(x) is negative, the graph is concave down.

Since both f'(x) and f''(x) are less than 0 (negative), look for a point where the graph is decreasing and concave down.