Real Solutions of Exponential Equations ##e^x = x^2## & ##x^3##

[juantheron]

[1] Total no. of real solution of the equation $e^x = x^2$

[2] Total no. of real solution of the equation $e^x = x^3$

My Solution:: [1] Let $f(x) = e^x$ and $g(x) = x^2$

Now we have use Camparasion Test for derivative

So $f^{'}(x) = e^x$ which is $>0\forall x\in \mathbb{R}$ and $g^{'}(x) = 2x$

So When $x<0$. Then $f(x)$ is Increasing function and $g(x)$ is Decreasing function

So exactly one solution for $x\leq 0$

Now for $x\geq 1$. Then $f(x)$ is Increasing faster then $g(x)$ . So here curve does not Intersect

Now we will check for $0<x<1$

I Did not understand have can i check here which one is Increasing faster

so please help me

Thanks

 

[jbunniii]

Hint: What are the minimum and maximum values of $f$ and $g$ in the interval $[0,1]$? Where do those values occur?

 

[juantheron]

Thanks jbunniii Got it

Here we have to calculate which curve is above and which is below in the Interval $\left (0,1 \right)$

Given $e^x = x^2 \Rightarrow e^x - x^2 = \underbrace{\left(e^x - 1\right)}_{ > 0}+\underbrace{\left(1 - x^2\right)}_{ > 0} > 0\; \forall x\in \left(0,1\right)$

So $e^x - x^2 >0\Rightarrow e^x > x^2$ in $x \in \left(0,1\right)$

So first equation has only Real Roots