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

  • Thread starter juantheron
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    Exponential
In summary, the equation ##e^x = x^2## has exactly one real solution in the interval ##x\leq 0## and no real solutions in the interval ##x>0##. This is determined by using the Comparison Test for derivatives and analyzing the behavior of the two functions ##f(x) = e^x## and ##g(x) = x^2## in different intervals.
  • #1
juantheron
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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
 
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  • #2
Hint: What are the minimum and maximum values of ##f## and ##g## in the interval ##[0,1]##? Where do those values occur?
 
  • #3
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
 

1. What are the real solutions of the exponential equation ex = x2?

The real solutions of this equation can be found by graphing the two functions ex and x2 and finding the points of intersection. These points represent the real values of x that satisfy the equation.

2. Can there be more than one real solution for the equation ex = x2?

Yes, there can be more than one real solution for this equation. In fact, there are two real solutions: x = 0 and x ≈ 0.703204.

3. How do you solve the equation ex = x3?

This equation does not have any real solutions. The only solution to this equation is a complex number, which cannot be represented on a traditional graph.

4. What is the significance of the real solutions for exponential equations?

The real solutions represent the points where the two functions ex and x2 intersect. These points can have practical applications in various scientific and mathematical fields, such as determining the growth rate of a population or solving optimization problems.

5. Can exponential equations with different bases have real solutions?

Yes, exponential equations with different bases can have real solutions. The process for finding the real solutions may differ depending on the specific equation, but they can be found by graphing or using algebraic methods.

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