Question

Can the Intermediate Value Theorem be applied to show that there is a root of the
equation X^5 - X³ + 3x - 5 =0 in the given interval [2,1] ? If yes, apply it

Answer 1

Define the function $f(x)=x^5-x^3+3x-5.$ The function $f(x)$ is continuous on the closed interval $[1, 2]$ as polynomial.

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

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

By the Intermediate Value Theorem $f$ must have a zero between $1$ and $2.$

Hence the Intermediate Value Theorem can be applied to show that there is a root of the

equation $x^5-x^3+3x-5=0$ in the given interval $[1,2].$