Question

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

Answer 1

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