The question in the image is a mathematical proof problem. It involves proving an inequality related to an integral function. Here's the summary:

We are given a continuous and differentiable function $f(x)$ on the interval $[a, b]$, with the condition:

$\int_a^b f(x) \, dx = 0$

We need to prove the inequality:

$\left( \int_a^{2b-a} f(x) \, dx \right)^2 \leq \frac{2(b - a)^3}{3} \int_a^{2b} (f'(x))^2 \, dx$

This appears to be a problem that combines integral calculus with inequality proving. The method of solution could involve the application of certain integral inequalities, possibly related to the Cauchy-Schwarz inequality or integration by parts.

Would you like me to go through the solution in detail?