Find a nonnegative function 
whose ordinate set on an interval has area proportional to the product of the function values at the endpoints of the interval.
With reference to this previous exercise (Section 8.24, Exercise #18) we write
![\[ A(x) = \int_a^x f(t) \, dt = C f(x) f(a) \quad \implies \quad \int_a^a f(t) \, dt = C(f(a))^2. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-72dd4e0ae857403576a7df913e0089c8_l3.png "Rendered by QuickLaTeX.com")
But
![\[ \int_a^a f(t) \, dt = 0 \quad \implies \quad C(f(a))^2 = 0 \quad \implies \quad C = 0 \text{ or } f(a) =0. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-54dcff21cb882938d99b39a5b3296bc3_l3.png "Rendered by QuickLaTeX.com")
As in the previous exercise (Section 8.24, Exercise #19) this implies
![\[ f(x) = 0 \qquad \text{for all } x. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-85c35577c2538b3fb27db5738c9c76e3_l3.png "Rendered by QuickLaTeX.com")