Integral Inequality There is a problem in my book that says:
Show that (assuming $f,g,f^2, g^2, $and$ fg$ are integrable) $$\int_{a}^{b}fg\leq \sqrt{\int_{a}^{b}f^2}\sqrt{\int_{a}^{b}g^2}$$.  I know this is the Cauchy-Shwarz Inequality for integrals, but my book says to prove it by defining $$p(\lambda)=\int_{a}^{b}[f-\lambda g]^2$$ for each real number $\lambda$, and then show that $p(\lambda)$ is a quadratic polynomial for which $p(\lambda)\geq0$, for all $\lambda$, and thus the discriminant is not positive. How does this help?

$$p(\lambda)=\lambda^2\int g^2 -2\lambda \int fg +\int f^2$$
Now what is $b^2-4ac$, in terms of the question you mention?  Cancel $\lambda^2$ from that being $\geq 0$...
