There is an important relation between the increment quotient \({\displaystyle \frac{\Delta y}{\Delta x}}\) and \(\dfrac{dy}{dx}=f^\prime(x)\). In calculus, many properties of functions can be deduced from this relation, which is called the **Mean-Value Theorem**. Let’s first consider the geometric significance of this theorem.

Consider the graph of \(y=f(x)\) and two points \(A(a,f(a))\) and \(B(b,f(b))\) on it (Figure 1). Because \(AG=b-a\) and \(GB=f(b)-f(a)\), the slope of the chord \(AB\) is \[\tan(\angle{GAB})=\frac{GB}{AG}=\frac{f(b)-f(a)}{b-a}=\frac{\Delta f}{\Delta x}.\tag{i}\]

As we can see, there is at least one point on the curve between \(A\) and \(B\) (as \(P\)) where the tangent to the curve \(y=f(x)\) is parallel to the chord \(AB\). If the \(x\)-value of \(P\) is \(c\), the slope at \(P\) is \[\tan\theta=f^\prime(c)=\tan(\angle{GAB}).\tag{ii}\] Equating (i) and (ii), we get \[\frac{f(b)-f(a)}{b-a}=f^\prime(c).\tag{iii}\]

In fact, we will find Eq. (iii) intuitive if we think of \(f(t)\) as the position of a particle (that moves on a straight line) at time \(t\). The left-hand side of Eq. (iii) represents the average velocity in the time interval \([a,b]\) and the right-hand side of Eq. (iii) is the instantaneous velocity at some time. This equation states that at some instant the instantaneous velocity must be equal to the average velocity. For example, if the average speed (or average velocity) of a train is 100 km/h (kilometers per hour), then its speedometer must register 100 km/h *at least* once during the trip.

The formal statement of the mean-value theorem is as follows.

**Theorem 1.The MEAN-VALUE THEOREM**** FOR DERIVATIVES**. If function \(f(x)\) is a function with the following properties:

*\(f\) is continuous on a closed interval \([a,b]\) and**\(f\)is differentiable on the open interval \((a,b)\),*

*then there exists at least one point \(c\) in the open interval \((a,b)\) such that \[f^\prime(c)=\frac{f(b)-f(a)}{b-a},\tag{iii}\] or equivalently, \[f(b)-f(a)=f^\prime(c)(b-a).\tag{iv}\]*

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- The Mean-Value Theorem for derivatives is sometimes called Lagrange’s Mean-Value Theorem.
- Note that the Mean-Value Theorem does not assert where the mean value \(c\) is located, except that it is somewhere between \(a\) and \(b\). In fact, except for special cases, it is often difficult to determine this point. However, many important properties can be deduced from the mere existence of such a point.
- Similar to Rolle’s theorem where \(f^\prime\) vanishes, there might be more than one point that satisfies the Mean-Value Theorem. For example, see Figure 3 where two points satisfy Eq. (iii).
- If at any point in the interval \((a,b)\), the derivative \(f^\prime\) fails to exists, the Mean-Value Theorem may not hold true.

We can express Eq. (iii) somewhat differently by noticing that the number \(c\) can be written as \[c=a+\theta(b-a),\] where \(\theta\) is a certain number between 0 and 1 (\(0<\theta<1\)). Now if we replace \(a\) by \(x\) and \(b\) by \(x+h\), we can express the Mean-Value Theorem by the formula \[\frac{f(x+h)-f(x)}{h}=f^\prime(x+\theta h),\] or \[f(x+h)=f(x)+hf^\prime(x+\theta h)\qquad\text{where }0<\theta<1.\]

**Theorem 2. ****CAUCHY’S MEAN-VALUE FORMULA****.** Assume \(f(x)\) and \(g(x)\) are continuous on \([a,b]\) and are differentiable on \((a,b)\). Then there exists at least one point \(c\) in \((a,b)\) such that \[f^\prime(c)\left[g(b)-g(a)\right]=g^\prime(c)\left[f(b)-f(a)\right].\] If \(g^\prime(c)\neq0\) and \(g(a)\neq g(b)\), this is equivalent to: *\[\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f^\prime(c)}{g^\prime(c)}.\]*

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- Notice that the Mean-Value Theorem (Lagrange’s theorem) is the special case of Cauchy’s Mean-Value Formula when \(g(x)=x\).