Calculus I · Unit 2B · review

Derivative Theorems and Shape Review

Review

Summary

Critical numbers locate candidates. The closed interval method finds absolute extrema. Rolle's and Mean Value Theorems guarantee interior derivative behavior under continuity and differentiability hypotheses. Signs of $f'$ determine monotonicity; signs of $f"$ determine concavity.

Exercise

Find absolute extrema of $x^4-4x^2$ on $[-3,2]$ .

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Exercise 1 answer

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Exercise

Verify Rolle's Theorem for $f(x)=\sin x$ on $[0,\pi]$ .

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Exercise 2 answer

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Exercise

Apply the Mean Value Theorem to $f(x)=\sqrt x$ on $[1,4]$ .

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Exercise 3 answer

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Exercise

Analyze increasing and decreasing intervals of $x^4-4x^3$ .

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Exercise 4 answer

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Exercise

Classify all critical points of $x^4-4x^2$ .

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Exercise 5 answer

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Exercise

Find concavity and inflection points of $x^3-6x^2+9x$ .

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Exercise 6 answer

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Exercise

Perform a complete curve analysis of $(x^2-1)/(x^2+1)$ .

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Exercise 7 answer

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Exercise

Explain why $f"(c)=0$ is not sufficient for an inflection point.

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Exercise 8 answer

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After the explanation

Use the section idea

Reading lens

Turn derivative signs and theorem hypotheses into a defensible account of extrema, monotonicity, concavity, and global shape.

Mental model

Critical numbers divide the domain into testable intervals; endpoints and discontinuities keep local evidence from becoming an unjustified global claim.

Decision

List the domain and candidates, test derivative signs, compare endpoint values, and verify each theorem's hypotheses explicitly.

Common trap

Calling every point with f-prime zero an extremum or every point with f-double-prime zero an inflection point.

Check yourself

Can every turn, bend, endpoint result, and asymptote in your sketch be traced to algebraic evidence?

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