For what values of x does the graph of #f(x)=e^(3x)# and #g(x)=x^2-2# have a horizontal tangent line?

1 Answer
Feb 25, 2015

The graph of #f(x) = e^(3x)# does not have a horizontal tangent line.

The graph of #g(x) = x^2 -2# has a tangent line at #x=0#

The two functions do not share a tangent line (which seems to be what you are asking for).

Only read past this point to see where I got the tangent line value (for #g(x)#) and the non-existence of a tangent line (for #f(x)#):

"horizontal tangent line" #rarr# slope #= 0#

For #g(x)# (always do the easy one first)
the slope at #x# is given by the derivative of #g(x)#
#g'(x) = 2x#

for a slope of #0#
#g'(x) = 2x = 0#
#rarr x = 0#

For #f(x) = e^(3x)#
#f'(x) = (d (3x))/dx * e^(3x)#
#f'(x) = 3 e^(3x)#

but
#e^(anynumber)# is always #>0#
so
#f(x)# does not have a horizontal tangent