Sử dụng Định lý Giá trị Trung bình

Somewhere, the instant matches the average

• Drive $120$ km in $2$ hours: your average speed is $60\,\tfrac{\text{km}}{\text{h}}$.
• At some instant your speedometer must have read exactly $60$ — you can't average $60$ without hitting it.
• That intuitive fact is the Mean Value Theorem 中值定理 (MVT).
• It links the average rate of change over an interval to an instantaneous rate somewhere inside.

The hypotheses come first

• MVT needs $f$ continuous on $[a,b]$ and differentiable on $(a,b)$.
• If both hold, there is at least one $c$ in $(a,b)$ with
• $$f'(c)=\frac{f(b)-f(a)}{b-a}$$
• The left side is an instantaneous rate; the right side is the average rate over $[a,b]$.

The Mean Value Theorem: some tangent is parallel to the secant over the interval

A tangent parallel to the secant

• The right-hand side is the slope of the secant line joining the endpoints.
• So MVT guarantees a point where the tangent line is parallel to that secant.
• Geometrically: somewhere the curve's slope equals the overall average slope.
• Like the IVT, it is an existence theorem — it promises $c$ exists, not its value.

Finding the guaranteed $c$

• To find $c$: set $f'(c)$ equal to the average rate and solve for $c$ in $(a,b)$.
• First check the hypotheses (continuous + differentiable) — skip that and the theorem doesn't apply.
• Discard any solution outside the open interval $(a,b)$.
• The MVT is the engine behind many later results (like "if $f'=0$ everywhere, $f$ is constant").