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EEJ MAIN Mathematics QUESTION #6009
Question 1

Let $f:[0,1]\to\mathbb{R}$ satisfy $f(xy)=f(x)\cdot f(y)$ for all $x,y\in[0,1]$ with $f(0)\neq 0$. If $y=y(x)$ solves $\dfrac{dy}{dx} = f(x)$ with $y(0)=1$, find $y\!\left(\dfrac{1}{4}\right)+y\!\left(\dfrac{3}{4}\right)$.

  • 3✔️
  • 4
  • 2
  • 5
Correct Answer Explanation

From the functional equation $f(xy)=f(x)f(y)$ with $f(0)\neq 0$, putting $x=y=0$ gives $f(0)=f(0)^2 \Rightarrow f(0)=1$. Putting $y=0$: $f(0)=f(x)f(0) \Rightarrow f(x)=1$ for all $x$.

So $\frac{dy}{dx}=1$, giving $y=x+C$. With $y(0)=1$: $C=1$, so $y=x+1$.

$y(1/4)+y(3/4)=(1/4+1)+(3/4+1)=5/4+7/4=12/4=\mathbf{3}$