Question

Verify mean value theorem for the function $f\left(x\right)=x^2$ in the interval [2,4].

Answer 1

$f\left(x\right)=x^2$ in interval [2,4].

Now, checking the conditions for mean value theorem

Since, f(x) is a polynomial, it is continuous in (2,4).

Also, it is differentiable in (2,4).

Now, $a=2$ and $b=4$

$f\left(a\right)=f\left(2\right)=4$

$f\left(b\right)=f\left(4\right)=16$

$f\left(x\right)=x^2$

$f^{\prime }\left(x\right)=2x$

$f^{\prime }\left(c\right)=2c$

Now,

$f^{\prime }\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

$2c=\frac{16-4}{4-2}$

$2c=6$

$c=3$

$c=3\in (2,4)$

Hence, Mean value theorem is satisfied.