Which condition identifies a location of a relative maximum for a differentiable function?

Which condition identifies a location of a relative maximum for a differentiable function?

• A. f'(c) = 0 OR f'(x) is undefined AND f'(c) switches from positive to negative
• B. f'(c) = 0
• C. f''(c) < 0
• D. f'(c) ≠ 0

Answer: (A)

The main idea is the first derivative test for a local maximum: a point where the function goes up as you approach from the left and goes down as you leave to the right. For a differentiable function, that shows up as the derivative being positive just before the point and negative just after it, so the tangent slope is zero at the peak. But a local maximum can also occur if the derivative does not exist at the point, provided the left-hand slope is positive and the right-hand slope is negative, meaning the function increases then decreases around that point. That’s why the most complete condition is that at the candidate location the derivative is zero, or the derivative is undefined with a sign change from positive to negative across the point. If you only had the derivative equal to zero, that isn’t enough by itself, since a horizontal tangent can occur at an inflection point or a non-maximum. Relying on the second derivative test, f''(c) < 0 is a sufficient condition when f'(c)=0 and the function is twice differentiable, but not necessary and doesn’t cover cases where the second derivative doesn’t exist or f'(c) = 0 but the sign change isn’t clear. If the derivative is nonzero at the point, the function is still increasing or decreasing there, so it cannot be a local maximum.

The main idea is the first derivative test for a local maximum: a point where the function goes up as you approach from the left and goes down as you leave to the right. For a differentiable function, that shows up as the derivative being positive just before the point and negative just after it, so the tangent slope is zero at the peak. But a local maximum can also occur if the derivative does not exist at the point, provided the left-hand slope is positive and the right-hand slope is negative, meaning the function increases then decreases around that point. That’s why the most complete condition is that at the candidate location the derivative is zero, or the derivative is undefined with a sign change from positive to negative across the point.

If you only had the derivative equal to zero, that isn’t enough by itself, since a horizontal tangent can occur at an inflection point or a non-maximum. Relying on the second derivative test, f''(c) < 0 is a sufficient condition when f'(c)=0 and the function is twice differentiable, but not necessary and doesn’t cover cases where the second derivative doesn’t exist or f'(c) = 0 but the sign change isn’t clear. If the derivative is nonzero at the point, the function is still increasing or decreasing there, so it cannot be a local maximum.