Answer
\begin{align*}
f(x) = 36x^{3}+9x-43
\end{align*}
Work Step by Step
It is given that the curve has the following second derivative:
\begin{align*}
\frac{d^{2}x}{dx^{2}} = 6x
\end{align*}
Also, it is given that the curve y = 5 - 3x is tangent to the curve at point x = 1. We will use this piece of information in two ways:
(1) y(1) = 5 - 3$\times$1 = 2. This means that both curves pass through the point (1, 2).
(2) y' = -3. This means that the slope of both curves is -3 at x = 1.
Thus, let us find integrate the second derivative and use (2):
\begin{alignat}{3}
\int 6xdx=&12x^{2}+C = -3 \\
& 12\times1^{2} + C = -3 \\
& C = 9 \\
\end{alignat}
Finally, we can integrate one more time and use (1):
\begin{alignat}{3}
\int(12x^{2}+9)&dx=36x^{3}+9x+C \\
& 36\times1^{3}+9\times1+C=2 \\
& C = -43 \\
\end{alignat}
Thus, the final answer is
\begin{align*}
f(x) = 36x^{3}+9x-43
\end{align*}
