A point c is a relative minimum if which condition holds?

A point c is a relative minimum if which condition holds?

• A. f'(c) > 0
• B. f'(c) undefined
• C. f'(c) < 0
• D. f'(c) = 0 AND f' changes from negative to positive

Answer: (D)

A relative minimum occurs where the function changes from decreasing to increasing, so the derivative crosses from negative to positive at that point. The given condition requires the slope to be zero at c (a stationary point) and also for the derivative to change from negative to positive as you pass c. That combination guarantees the function was decreasing up to c and then increases after c, which is what a local minimum is. If the derivative were merely positive at c, the function would be rising there, not at a minimum. If f′(c) is undefined, you could have a cusp or vertical tangent, which doesn't by itself ensure a local minimum. If f′(c) < 0, the function is still decreasing at c, not a minimum.

A relative minimum occurs where the function changes from decreasing to increasing, so the derivative crosses from negative to positive at that point. The given condition requires the slope to be zero at c (a stationary point) and also for the derivative to change from negative to positive as you pass c. That combination guarantees the function was decreasing up to c and then increases after c, which is what a local minimum is.

If the derivative were merely positive at c, the function would be rising there, not at a minimum. If f′(c) is undefined, you could have a cusp or vertical tangent, which doesn't by itself ensure a local minimum. If f′(c) < 0, the function is still decreasing at c, not a minimum.