Find the particular solution determined by the initial condition.
\[
f^{\prime}(x)=4 x^{3 / 4}-x^{2} ; f(1)=-9
\]
\[
f(x)=
\]

Step 1 ：We are given the first order differential equation \(f'(x) = 4x^{3/4} - x^2\) with the initial condition \(f(1) = -9\).

Step 2 ：To find the particular solution, we first need to find the general solution by integrating the function.

Step 3 ：The general solution of the differential equation is \(f(x) = C1 + 2.28571428571429x^{7/4} - 0.333333333333333x^3\), where \(C1\) is the constant of integration.

Step 4 ：We can find the particular solution by substitifying the initial condition into the general solution.

Step 5 ：Substituting \(x = 1\) and \(f(1) = -9\) into the general solution, we get \(-9 = C1 + 1.95238095238095\).

Step 6 ：Solving for \(C1\), we get \(C1 = -10.9523809523810\).

Step 7 ：Substituting \(C1\) back into the general solution, we get the particular solution \(f(x) = 2.28571428571429x^{7/4} - 0.333333333333333x^3 - 10.952380952381\).

Step 8 ：\(\boxed{f(x) = 2.28571428571429x^{7/4} - 0.333333333333333x^3 - 10.952380952381}\) is the particular solution of the given differential equation with the initial condition \(f(1) = -9\).