Maxima and Minima

Extrema and the extreme value theorem

Critical points and the closed interval method

The first derivative test

The second derivative test

Fermat's theorem on local extrema applies to an interior point c where f is differentiable and has a local extremum. What conclusion does it draw?

Which points belong on the critical point list of a function f?

To find the absolute extrema of a continuous function on [a, b], which values must be compared?

At a critical point c, the derivative f prime changes from positive to negative as x passes through c. What does the first derivative test conclude?

The second derivative test examines f double prime of c at a stationary point c. What does f double prime of c = 0 tell us?

The function f(x) = x on the open interval (0, 1) has no maximum or minimum there. Which hypothesis of the extreme value theorem fails?

The function f(x) = x^3 - 3x has derivative 3x^2 - 3. Besides x = 1, what is its other critical point? Answer with a number.

On the interval [0, 2], the values of f(x) = x^3 - 3x at the candidates are f(0) = 0, f(1) = -2, and f(2) = 2. What is the absolute minimum value? Answer with a number.

The function f(x) = x^4 - 2x^2 has derivative 4x(x^2 - 1). How many stationary points does it have? Answer with a number.

For f(x) = x^4, both f prime and f double prime vanish at c = 0, so the second derivative test is inconclusive, yet the point is a local minimum. How many local extrema does x^4 have on the whole real line? Answer with a number.