- #1
juantheron
- 247
- 1
[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
[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