Let f:[0,1]R satisfy f(xy)=f(x)f(y), f(0)0.

Question

Let $f:[0,1]	o\mathbb{R}$ satisfy $f(xy)=f(x)f(y)$, $f(0)
e0$. If $y=y(x)$ solves $\dfrac{dy}{dx}=f(x)$, $y(0)=1$, then $y\left(\dfrac14ight)+y\left(\dfrac34ight)$ is:

Accepted Answer

The functional equation forces $f\equiv1$. So $y=x+1$, and $y(1/4)+y(3/4) = 5/4+7/4 = 3$.

Solutions · 1