Find a function $f(x)$ such that $f^{\prime}(x)=9 e^{x}-9 x$ and $f(0)=-5$ \[ f(x)= \]

Step 1 ：Given that $f^{\prime}(x)=9 e^{x}-9 x$, we can integrate both sides with respect to $x$ to find $f(x)$.

Step 2 ：The integral of $f^{\prime}(x)$ with respect to $x$ is $f(x)$, and the integral of $9 e^{x}-9 x$ with respect to $x$ is $9 e^{x}-\frac{9}{2} x^{2}+C$, where $C$ is the constant of integration.

Step 3 ：So, $f(x)=9 e^{x}-\frac{9}{2} x^{2}+C$.

Step 4 ：We are also given that $f(0)=-5$. Substituting $x=0$ into the equation gives $-5=9+C$.

Step 5 ：Solving for $C$ gives $C=-14$.

Step 6 ：Substituting $C=-14$ back into the equation for $f(x)$ gives $f(x)=9 e^{x}-\frac{9}{2} x^{2}-14$.

Step 7 ：So, the function $f(x)$ that satisfies the given conditions is $f(x)=9 e^{x}-\frac{9}{2} x^{2}-14$.

Step 8 ：Checking the solution, we find that $f^{\prime}(x)=9 e^{x}-9 x$ and $f(0)=-5$, which are the conditions given in the problem. Therefore, the solution is correct.