Mathematics · Ch 9 — Applications of Derivatives

Lagrange's Mean Value Theorem (LMVT)

9.3.2

Lagrange's Mean Value Theorem (LMVT)

Lagrange's Mean Value Theorem (LMVT): if a real-valued function $f$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$ , then there exists at least one point $c$ in $(a,b)$ such that

f'(c) = \frac{f(b)-f(a)}{b-a}

In words: for any differentiable function that is continuous at its two endpoints, there is at least one point in between where the instantaneous rate of change (the tangent's slope) equals the average rate of change over the whole interval (the chord's slope). LMVT is a direct generalisation of Rolle's Theorem — Rolle's is the special case where $f(a)=f(b)$ , making the right-hand side zero.

Geometrical significance. Draw the curve $y=f(x)$ and mark its endpoints $A(a,f(a))$ and $B(b,f(b))$ . The slope of the chord $AB$ is $\dfrac{f(b)-f(a)}{b-a}$ . By the statement of LMVT, $f'(c)$ equals this chord slope for some $c\in(a,b)$ — meaning the tangent to the curve at $x=c$ is parallel to the chord $AB$ .

Figure 2.3.2Fig. 2.3.2 — geometrical significance of LMVT: a curve with chord AB from A(a, f(a)) to B(b, f(b)) and a tangent at C(c) parallel to the chord.

$Fig. 2.3.2 — Fig. 2.3.2 — geometrical significance of LMVT: a curve with chord AB from A(a, f(a)) to B(b, f(b)) and a tangent at C(c) parallel to the chord.$

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. This figure illustrates the geometric meaning of Lagrange's Mean Value Theorem. The curve $y=f(x)$ is drawn over $[a,b]$ with its two endpoints marked $A(a,f(a))$ and $B(b,f(b))$ , and a straight chord (secant line) is drawn joining $A$ and $B$ . Somewhere on the curve between $A$ and $B$ , the figure marks a point $x=c$ where the tangent line to the curve is drawn parallel to this chord, illustrating that the instantaneous slope $f'(c)$ …

Worked Example 1. Verify LMVT for $f(x)=\sqrt{x+4}$ on $[0,5]$ . The function is continuous on $[0,5]$ and differentiable on $(0,5)$ , so LMVT applies. $f'(x)=\dfrac{1}{2\sqrt{x+4}}$ . $f(0)=\sqrt4=2$ , $f(5)=\sqrt9=3$ . Setting $f'(c)$ equal to the chord slope: $\dfrac{1}{2\sqrt{c+4}}=\dfrac{3-2}{5-0}=\dfrac15$ , so $\sqrt{c+4}=\dfrac52$ , giving $c+4=\dfrac{25}{4}$ , so $c=\dfrac94$ , which lies in $(0,5)$ — LMVT is verified. …

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