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Exponential equation

  • Thread starter juantheron
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  • #1
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[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
 

Answers and Replies

  • #2
jbunniii
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Hint: What are the minimum and maximum values of ##f## and ##g## in the interval ##[0,1]##? Where do those values occur?
 
  • #3
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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
 

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