Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 3 / d / Solution

The supplied derivative identity and the Cauchy-Schwarz inequality giveThereforeThe sum $W_t+B_{1-t}$ is standard normal for every $t$, so the right side is $2\mathbb{E}{f}^{\prime }(W_1{)}^2$. Part c proves the Gaussian logarithmic Sobolev inequality