Let ${f : [0, 1] \rightarrow [-1, 1] }$ is a continuous function such that ${ \int_{0}^{1} x f \left(x\right) dx =0}$
Find $f(x)$ such that ${ \int_{0}^{1} \left(x ^{2 } + \frac{1}{4} \right) f \left(x\right) dx}$ has the maximum value.
Let ${f : [0, 1] \rightarrow [-1, 1] }$ is a continuous function such that ${ \int_{0}^{1} x f \left(x\right) dx =0}$
Find $f(x)$ such that ${ \int_{0}^{1} \left(x ^{2 } + \frac{1}{4} \right) f \left(x\right) dx}$ has the maximum value.