Set $f'(c)$ equal to the average rate of change, then solve for $c$
Discard any solutions for $c$ that do not lie strictly inside the open interval $(a,b)$
4. 应用中值定理证明函数行为和解决问题★★★☆☆⏱ 3 min
Beyond routine calculation, MVT is used to justify higher-order conclusions about function behavior, a common FRQ skill. If you know $m \leq f'(x) \leq M$ for all $x$ in $[a,b]$, MVT tells you $m(b-a) \leq f(b) - f(a) \leq M(b-a)$. This is also the theoretical foundation for the rule that a positive derivative everywhere on an interval implies the function is increasing on that interval.
Common Pitfalls
Why: Students mix up interval requirements because derivatives are rarely discussed at endpoints.
Why: Students only check continuity and forget that non-differentiability at an interior point violates the second hypothesis.
Why: Students misremember MVT as guaranteeing $c$ in $[a,b]$ instead of $(a,b)$.
Why: Students misread 'at least one' as 'exactly one'.
Why: Students focus on the extra $f(a)=f(b)$ condition and forget to check core hypotheses first.