Let \(f:\mathbb{R}\rightarrow\mathbb{R}\) be an odd function and
\[
\int_{-1}^{1}x^3f''(x)\,dx=\frac85.
\]
If
\[
g(x)=xf(x),
\]
\[
g'(1)=\frac83
\]
and
\[
\int_0^1g(x)\,dx=\frac{2}{15},
\]
then \(f(1)=\)
Show Hint
For odd functions,
\[
\boxed{f(-x)=-f(x),\qquad
f''(-x)=-f''(x).}
\]
Use symmetry first, then apply integration by parts to simplify integrals involving derivatives.