The marginal cost function is given by
\[
C'(x) = x^{2/5} + 3.
\]
To find the cost function \( C(x) \), we integrate the marginal cost function:
\[
C(x) = \int (x^{2/5} + 3) \, dx.
\]
Calculating the integral, we have:
\[
C(x) = 3x + \frac{x^{\frac{2}{5} + 1}}{\frac{2}{5} + 1} + C,
\]
which simplifies to:
\[
C(x) = C + \frac{5}{7} x^{7/5} + 3x.
\]
We are given that the cost of producing 32 units is \$252, which provides the initial condition:
\[
C(32) = 252.
\]
Substituting \( x = 32 \) into the general solution:
\[
252 = C + \frac{5}{7} (32)^{7/5} + 3(32).
\]
Calculating \( (32)^{7/5} \):
\[
(32)^{7/5} = 32^{1.4} = 32^{\frac{7}{5}} = 2^{7} = 128.
\]
Now substituting this value back into the equation:
\[
252 = C + \frac{5}{7} \cdot 128 + 96.
\]
Calculating \( \frac{5}{7} \cdot 128 \):
\[
\frac{5 \cdot 128}{7} = \frac{640}{7}.
\]
Thus, we have:
\[
252 = C + \frac{640}{7} + 96.
\]
Converting 96 to a fraction with a denominator of 7:
\[
96 = \frac{672}{7}.
\]
Now, substituting this into the equation:
\[
252 = C + \frac{640 + 672}{7} = C + \frac{1312}{7}.
\]
To isolate \( C \):
\[
C = 252 - \frac{1312}{7}.
\]
Converting 252 to a fraction:
\[
252 = \frac{1764}{7}.
\]
Thus:
\[
C = \frac{1764 - 1312}{7} = \frac{452}{7}.
\]
Now substituting \( C \) back into the general solution, we have the particular solution:
\[
C(x) = \frac{452}{7} + \frac{5}{7} x^{7/5} + 3x.
\]
The cost function is
\[
\boxed{C(x) = \frac{452}{7} + \frac{5}{7} x^{7/5} + 3x}.
\]
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