Cauchy Euler Mean Value Theorem Engineering Maths, Btech first year

Cauchy Mean Value Theorem

Let $f(x)$ and $g(x)$ be two functions which are both:
a) differentiable in [a,b]
b) $g^{\prime }(x)\ne 0$ for any value of $x$ in [a,b]
then there exists at least one value $c$ in between $a$ and $b$ such that $$\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f^{\prime }(c)}{g^{\prime }(c)}$$

Sample Problem

Verify Cauchy Mean Value Theorem for $$f(x)=x^4,g(x)=x^2\text{ in }[a,b]$$

Solution

$$f(x)=x^4,g(x)=x^2$$ $$f^{\prime }(x)=4x^3,g^{\prime }(x)=2x$$ By Cauchy Mean Value Theorem, $$\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f^{\prime }(c)}{g^{\prime }(c)}$$ $$\frac{b^4-a^4}{b^2-a^2}=\frac{4c^3}{2c}$$ $$2c^2=\frac{\cancel{(b^2-a^2)}(b^2+a^2)}{\cancel{(b^2-a^2)}}$$ $$c=\pm \frac{1}{\sqrt{2}}\sqrt{b^2+a^2}$$ $$\text{Since }c=-\frac{1}{\sqrt{2}}\sqrt{b^2+a^2}\notin [a,b]$$ $$\boxed{{\displaystyle c=\frac{1}{\sqrt{2}}\sqrt{b^2+a^2}}}$$