Answer to Question #156927 in Calculus for Koochika

Question #156927

Can the intermediate value theorem can be applied to show the that there is a root of equation x^5-x^3+3x-5 in the given interval [1,2] if yes, apply it


1
Expert's answer
2021-01-21T14:58:39-0500

The function "f(x)=x^5-x^3+3x-5" is continuus on "\\R" as a polynomial.

Then the function "f" is continuous on the closed interval "[1,2]."


"f(1)=(1)^5-(1)^3+3(1)-5=-2<0"


"f(2)=(2)^5-(2)^3+3(2)-5=25>0"




"f(1)=-2<0<25<f(2)"

Hence the Intermediate Value Theorem can be applied.

Then there exists a number "c" in "(0, 1)" such that "f(c)=0."

Therefore the equation "x^5-x^3+3x-5=0" has at least one root "c" in the interval "(1, 2)."





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