Show that the equation x^3=5 has a solution on the interval [1,2]

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- May 3rd 2009, 07:09 PMLivingg90Help applying the intermediate value theorem
Show that the equation x^3=5 has a solution on the interval [1,2]

- May 3rd 2009, 07:14 PMTheEmptySet
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