equal slopes of tangent lines for f(x) and g(x)

Question

 

f(x) = 2e^3x 

 

g(x) = 6x³

 

 

When are the slopes of the tangent lines of these two functions equal ?

 

Please explain well.

THANKS !

Answer 1

First, slope is nothing but the first derivative of the function. So if you want to get the slope of the tangent line drawn to the function f(x) at x=6, you need to calculate f'(x) first and then plug in x=6 there, so essentially computing f'(x=6).

 

Okay, so now we are given two functions. Taking derivatives would give us the general expression for slopes of the tangent lines with the functions f(x) and g(x). Since we want to determine the specific value at which the slopes are equal to each other, we need set the slopes' expressions equal to each other and solve for that specific x.

 

f'(x) = 6 e^3x

 

g'(x) = 18x²

 

6 e^3x = 18 x²

e^3x = 3x²

e^3x - 3x² = 0

 

I used a graphing tool to plot the function on the left side at it is zero at x = -0.344. So at x=-0.344 is where the two functions have equal slopes. Link to the plot:

 

https://www.desmos.com/screenshot/irkheoovgn