Question

3. Find an interval on which Rolle’s theorem applies to f(x) = x 3 − 7x 2 + 10x. Then, find all the critical points in that interval.

Answer 1

$f(x) = x^ 3 − 7x^ 2 + 10x$

$=x(x^2-7x+10)=x(x-2)(x-5)$

The interval for which The rolles theorem is applicable will be [0,5].

The critical point in that interval can be calculated by putting $f'(x)=0$

$3x^2-14x+10=0\\$

$x=\dfrac{14\pm \sqrt{196-120}}{6}=\dfrac{14\pm \sqrt{76}}{6}$

Hence the critical points are $\dfrac{14- \sqrt{76}}{6},\dfrac{14+ \sqrt{76}}{6}$