Which statement about absolute extrema is true?

Which statement about absolute extrema is true?

• A. They can only occur at endpoints.
• B. They must occur where f' = 0.
• C. They can occur at endpoints or at critical points where f' is zero or undefined.
• D. They cannot occur if the function is increasing.

Answer: (C)

Absolute extrema on a closed interval occur either at the endpoints or at interior points where the derivative is zero or undefined. If an interior point is an absolute maximum or minimum and the derivative exists there, Fermat’s principle says the slope must be zero. If the derivative isn’t defined at a point (such as a cusp or corner), that point can also be an extremum. The endpoints must be checked separately because the derivative test applies only to interior points. For example, the absolute maximum of f(x)=x^2 on [-1,2] occurs at x=2 (an endpoint), while the absolute minimum occurs at x=0 (an interior point where f' = 0). In another case like f(x)=|x| on [-1,1], the minimum occurs at the interior point x=0 where the derivative is undefined, while the maximum occurs at the endpoints. So the correct statement is that absolute extrema can occur at endpoints or at interior critical points where f' is zero or undefined. The other ideas are too restrictive or incomplete: extremes need not be only at endpoints, they can occur where the derivative is undefined, and an increasing function can still have an endpoint as an extremum.

Absolute extrema on a closed interval occur either at the endpoints or at interior points where the derivative is zero or undefined. If an interior point is an absolute maximum or minimum and the derivative exists there, Fermat’s principle says the slope must be zero. If the derivative isn’t defined at a point (such as a cusp or corner), that point can also be an extremum. The endpoints must be checked separately because the derivative test applies only to interior points.

For example, the absolute maximum of f(x)=x^2 on [-1,2] occurs at x=2 (an endpoint), while the absolute minimum occurs at x=0 (an interior point where f' = 0). In another case like f(x)=|x| on [-1,1], the minimum occurs at the interior point x=0 where the derivative is undefined, while the maximum occurs at the endpoints.

So the correct statement is that absolute extrema can occur at endpoints or at interior critical points where f' is zero or undefined. The other ideas are too restrictive or incomplete: extremes need not be only at endpoints, they can occur where the derivative is undefined, and an increasing function can still have an endpoint as an extremum.