hello there
in order to compute the tangent line u need
the value of the function f(2g(x)) at x=3
and the value of the derivative of the function
at the same point meaning F'(3)
first u have to find f(2g(x))
just substitute in f(x) 2g(x)
u get
now what we need is actually
to know the value of this expression we need
to know the value of g(3)
since g(x) is the inverse of f(x)
then f(g(3))=3
in other words:
by guessing we get that the solution is:
g(3)=1
(there is another solution but its hard to get
it actually we need approximation)
now that we have g(3)=1
then F(3)=16*1^4+8*1^3+1=25
NOW we have to know the value of
F'(3)(the derivative)
to now the value of F'(3) we have to
know the value of g'(3) since we already know the value
of g(3)
now since f(g(x)=x
if we derive both sides using the chain rule
we get
now divide both sides by f'(g(x))
now just subsitute x=3
u get
g'(3)=1/7
can u proceed from here??