Solve using intermediate value theorem: Show that 5^x-6 x=0 has a… - Vidyadip

Question

Solve using intermediate value theorem:
Show that $5^x-6 x=0$ has a root in $[1,2]$

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Answer

Let $f(x)=5^x-6 x$
$5^x$ and $6 x$ are continuous functins for all $x \in R$.
$\therefore 5^{ x }-6 x$ is also continuous for all $x \in R$.
i.e. $f(x)$ is continuous for all $x \in R$.
A root of $f(x)$ exists if $f(x)=0$ for at least one value of $x$.
$ f(1)=5^1-6(1)$
$=-1<0$
$f(2)=(5)^2-6(2)$
$=13>0$
$\therefore f(1)<0 \text { and } f(2)>0 $
By intermediate value theorem, there has to be a point ' $c$ ' between $1$ and $2$ such that $f(c)=0$.
$\therefore$ There is a root of the given equation in $[1,2]$.

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