Show that the equation x^3=5 has a solution on the interval [1,2]
Consider the function
$\displaystyle f(x)=x^3-5$ and note that
$\displaystyle f(1)=1^3-5=-4$ and
$\displaystyle f(2)=2^3-5=3$
Since polynomials are continous by the intermediate value theorem there exists $\displaystyle c \in (1,2)$ such that $\displaystyle f(c)=0$
$\displaystyle f(c)=0=c^3-5$ so the equation has a solution