Let $g:\mathbb{R}^2\to\mathbb{R}$ be a continuously differentiable function such that, for a given $y$, $g(x,y)$ does not change sign in the interval $[a,b]$ of $x$ and its derivatives are bounded.
Also, let $f:\mathbb{R}^2\to\mathbb{R}$ be a continuously differentiable function such that, for a given $y$, it attains a single maximum in the interior of the $x$-axis interval $[a,b]$, and its derivatives are bounded.
For $y = y_0$, definite integral of the product of these two functions over $[a,b]$ is zero, i.e., $$ \int_{a}^{b} f(x,y_0) g(x,y_0) dx = 0. $$
Furthermore, in the neighborhood of $y=y_0$, this integral takes the same sign as the difference $y-y_0$. That is, for example, when $y = y_+ > y_0$, the integral is positive, i.e., $$ \int_{a}^{b} f(x,y_+) g(x,y_+) dx > 0. $$
Can I infer anything about any norm of the function $f(x,y)$ locally in some neighborhood of $y=y_0$ from what is provided above?
P.S. This is my research and not a homework. The problem I'm asking here is a distilled down version of a bigger problem I'm stuck at. I've been pondering about this problem (tried using MVT for definite integrals and Holder's inequality to name a few things I've done) but couldn't find anything really useful.