Where are critical values located for a function with a differentiable domain?

Where are critical values located for a function with a differentiable domain?

• A. f''(c) = 0
• B. f'(c) = 0 or f'(x) is undefined
• C. f(c) = 0
• D. f'(c) > 0

Answer: (B)

Critical values occur where the slope of the function is zero or the slope fails to exist. If a point in the interior of the domain is a local maximum or minimum, and the derivative exists there, Fermat’s idea tells us the derivative must be zero at that point. If the derivative doesn’t exist at a point—such as at a sharp corner, a cusp, or a vertical tangent—that can also be a critical value because the function can have an extremum there or behave differently than the surrounding points. So the place to look for critical values is where f′(x) = 0 or where f′(x) is undefined. The other possibilities don’t capture that idea: a zero of f itself is a root, not a critical point, and a positive derivative means the function is increasing, not a point where the slope vanishes or breaks. The second derivative being zero is related to concavity, not to where the slope vanishes or ceases to exist.

Critical values occur where the slope of the function is zero or the slope fails to exist. If a point in the interior of the domain is a local maximum or minimum, and the derivative exists there, Fermat’s idea tells us the derivative must be zero at that point. If the derivative doesn’t exist at a point—such as at a sharp corner, a cusp, or a vertical tangent—that can also be a critical value because the function can have an extremum there or behave differently than the surrounding points.

So the place to look for critical values is where f′(x) = 0 or where f′(x) is undefined. The other possibilities don’t capture that idea: a zero of f itself is a root, not a critical point, and a positive derivative means the function is increasing, not a point where the slope vanishes or breaks. The second derivative being zero is related to concavity, not to where the slope vanishes or ceases to exist.