If f(x) is a continuous function on a closed interval x ∈ [a,b], then f(x) will have both an Absolute Maximum value and an Absolute Minimum value in the interval.

Where might extrema occur? The end points of a closed interval. The “critical numbers” for the function. Where do we find critical numbers? a. Where the derivative is equal to zero. b. Where the derivative does not exist. When will the derivative not exist? a. Discontinuity in the function b. A cusp or corner c. Vertical tangent line

Can the Extreme Value Theorem be applied to f(x)? f(x) is a polynomial and is continuous on all Real Numbers so it will be continuous on [-2,2]. The EVT is applicable. What are the AMax and the AMin for f(x) in [-2,2]? Continued 

Critical Points: f(x) has an AMax of 6 when x is 2 and f(x) has an AMin of -18 when x is -2

Mean Value Theorem for Derivatives If f (x) is continuous over [a,b] and differentiable over (a,b), then there is at least one value c between a and b such that: The Mean Value Theorem says that there is at least one point in the closed interval where the actual slope equals the average slope.

Can the Mean Value Theorem be applied to f(x)? f(x) is a polynomial and is continuous on all Real Numbers so it will be continuous on [-2,2]. f'(x) is a polynomial and is continuous on all Real Numbers so it will be continuous on (-2,2) so f(x) is differentiable on (-2,2). The MVT is applicable. Continued 

Find all values for c in (-2,2) guaranteed by the MVT. Not factorable. Use the quadratic formula to solve. Continued 

Both values for c are in the interval.

Rolle's Theorem If f (x) is continuous over [a,b], differentiable over (a,b), and f (a) = f (b) then there is at least one value c between a and b such that: Rolle's Theorem says that there is at least one point in the closed interval where there will be a local extreme value where c is a critical number for the function. Continued 

Critical numbers are x-values where the first derivative of the function is equal to zero or they are x-values where the first derivative of the function does not exist. If f(x) is continuous over [a,b] and f(a) = f(b)then there is at least one value c between a and b such that there will be a local extrema. Corollary to Rolle’s Theorem:

Can the Rolle's Theorem be applied to f(x)? f (x) is continuous over all Real Numbers so it is continuous over the indicated interval. ∴ Rolle's Theorem be applied to f(x). Continued 

Find all values c guaranteed by Rolle's Theorem. f '(x) is continuous over all Real Numbers so it is continuous over the indicated open interval and f (x) is differentiable over the open interval so there is at least one value c in the interval where f '(c) = 0.

Can the EVT, MVT, and/or Rolle's Theorem be applied to f(x)? If the theorem can be applied, find all that the theorem guarantees. If the theorem cannot be applied, explain why not. EVT: MVT: Rolle's Theorem: NO! f(x) is not continuous at x = 2.

Can the EVT, MVT, and/or Rolle's Theorem be applied to f(x)? If the theorem can be applied, find all that the theorem guarantees. If the theorem cannot be applied, explain why not. f(x) is continuous if x ≠ ± 2. EVT: ∴ f(x) is differentiable if x ≠ ± 2. Not in the interval! With no critical numbers in the interval, absolute extrema must exist at the endpoints. Continued 

MVT: c = Rolle's Theorem: Does not apply because f(3) ≠ f(6).

Increasing Function, Decreasing Function Precalculus Definition Calculus Definition

On which interval(s) is f(x) increasing or decreasing?

Antiderivative

Find g(x) which represents a specific antiderivative of f(x) that passes through

Connecting f '(x) and f ''(x) with the graph of f (x) What you were to have learned about:

First Derivative Test for Local Extrema

Concavity and Points of Inflection

Possible Inflection Point x =

Second Derivative Test for Local Extrema When the graph is concave down, the lines tangent to the points will be above the function. When the graph is concave up, the lines tangent to the points will be below the function.

CD LMax CU LMin