Unit 4 calc max and min

What is the extreme value theorem?

If f(x) is continuous, then there will a maximum and minimum at the candidate points

What does f’(x) > 0 mean?

f is increasing

What does f’(x) < 0 mean?

f is decreasing

What does f’’(x) > 0 mean?

f is concave up/ has a minimum

What does f’’(x) < 0 mean?

f is concave down/has a maximum

What does f’’(x) = 0 mean?

potential inflection (changes concavity)

What are critical points?

When y’ = 0 (slope is zero or DNE). Use them to find max and min on graphs

What is the First Derivative Test?

Finding derivatives to identify max, min, and points of inflection.

What are candidate points?

Absolute extrema (using critical points and end points) to find absolute max and absolute min

What is the second derivative test?

Finding the second derivative of a function to determine concavity ( f’’(x) > 0 concave up, f’’(x) < 0 concave down, f’’(x) = 0, potential inflection)

If a derivative goes from increasing to decreasing, what concavity is it?

concave down, maximum

If a derivative goes from decreasing to increasing, what concavity is it?

concave up, minimum