Interactive Real Analysis - part of MathCS.org

• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 6.1. Limits
• 6.2. Continuous Functions
• 6.3. Discontinuous Functions
• 6.4. Topology and Continuity
• 6.5. Differentiable Functions
• 6.6. A Function Primer
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

Theorem 6.5.12: Local Extrema and Monotonicity

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If f is differentiable on (a, b), and f has a local extrema at x = c, then f'(c) = 0.

If f'(x) > 0 on (a, b) then f is increasing on (a, b).

If f'(x) < 0 on (a, b) then f is decreasing on (a, b).

Context

Proof:

This proof is left as an exercise. For a hint on the first part you might want to look at the proof of Rolle's theorem. If you have proved the first part, it should be more or less clear how the remaining parts can be proved. Recall the definitions of an increasing or decreasing function and compare it with the difference quotient involved in the derivative of f.