Science:Math Exam Resources/Courses/MATH104/December 2012/Question 01 (o)/Solution 1

Define the function $f(x)$ as

$f(x)=5^{x}-10x-7.$

Notice that this function is continuous for all values of $x$. Hence, we need only find a closed interval where the sign of $f(x)$ changes and invoke the intermediate value theorem to get the desired conclusion.

Consider $x=2$ and $x=3$. If we take these two points and plug them into $f(x)$, we get

${\begin{aligned}f(2)&=5^{2}-10(2)-7=-2<0\\f(3)&=5^{3}-10(3)-7=88>0\end{aligned}}$

and so the sign of $f(x)$ changes over the closed interval $[2,3]$.

Therefore, there exists a value $c$ in $[2,3]$ such that

${\color {blue}f(c)=5^{c}-10c-7=0}$

by the Intermediate Value Theorem.