Answer to Question #181805 in Calculus for desmond

Question #181805

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.


1
Expert's answer
2021-04-20T07:48:07-0400

f(x)=x37x2+10xf(x) = x^ 3 − 7x^ 2 + 10x

=x(x27x+10)=x(x2)(x5)=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)=0f'(x)=0


3x214x+10=03x^2-14x+10=0\\

x=14±1961206=14±766x=\dfrac{14\pm \sqrt{196-120}}{6}=\dfrac{14\pm \sqrt{76}}{6}


Hence the critical points are 14766,14+766\dfrac{14- \sqrt{76}}{6},\dfrac{14+ \sqrt{76}}{6}


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