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C17 · Unit C5 Function analysis and optimization

Extreme Value and Mean Value Theorems

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 5.1–5.2
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

The Extreme Value Theorem guarantees an absolute minimum and maximum for a continuous function on a closed bounded interval. The Mean Value Theorem requires continuity on [a,b] and differentiability on (a,b), then guarantees c with f minutes (c) equals [f(b) minus f(a)]/(b minus a). Endpoints need not be differentiable. Rolle’s theorem is the equal-endpoint-value special case.

Worked example

For f(x) equals x squared on [1,3], the average slope is (9 minus 1)/2 equals 4. Since f minutes equals 2x, c equals 2 satisfies MVT. The polynomial meets both hypotheses. EVT also guarantees extrema, which here occur at endpoints.

Explore the mathematics

Extreme Value and Mean Value Theorems

Locate a tangent parallel to the interval secant under the Mean Value Theorem.

Setting 3 of 5

Scroll sideways to see the full diagram.

Predict where a tangent is parallel to the endpoint secant.xy-2-11234f(x)=x² on [0,2.5]

Predict before continuing

Predict where a tangent is parallel to the endpoint secant.

Read the complete explanation
  1. Predict where a tangent is parallel to the endpoint secant.
  2. The average rate is (b²−0)/(b−0)=b.
  3. Solve f′(c)=2c=b: c=b/2, strictly inside the interval.
  4. Here c=1.25. MVT requires continuity on the closed interval and differentiability on its interior.
See the idea: Extreme Value and Mean Value Theorems
See the idea: Extreme Value and Mean Value TheoremsFor x² on [1,3], the secant through (1,1) and (3,9) has slope 4. The tangent at (2,4) is parallel to it: f′(2)=4, illustrating MVT.xy101.52.5252.57.5310tangent at c=2

For x² on [1,3], the secant through (1,1) and (3,9) has slope 4. The tangent at (2,4) is parallel to it: f′(2)=4, illustrating MVT.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

A theorem’s conclusion may happen even when its hypotheses fail, but the theorem then provides no guarantee.

Investigate and explain

Create a function on an open interval with no attained maximum. Explain exactly which EVT hypothesis is missing.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.