use the intermidiate value theorem toshow that there is a root of the equation

Question

e^x=3x 

In the interval [0,1]

Answer 1

consider function f(x = e^x- 3x

it is continuous

f(0) = 1^x = 3x 

f(1) = e-3 which is negative.

 

This satisfies the conditions of the Intermediate Value Theorem, i.e. the continuous function transitions smoothly from positive to negative y values as x transits from 0 through 1.  Therefore at some point on [0,1] there has to be a point where y=0; at said x, the equation equation e^x = 3x is satisfied.