Question

Can the intermediate value theorem 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[?If yes , apply it

Answer 1

$f(x)= x^5-x^3+3x-5=0$

$f(1)=-2\\f(2)=25$

There is a change of sign of $f$ when $x$ passes from 1 to 2, it means there will be some intermediate value of $x$ in [1,2] for which $f(x) =0$ which means there is a root in [1,2].