Correct Answer - Option 1 : 3
The given equation is:
f(xy) = f(x)·f(y) for all x,y ∈ [0, 1] ….(i)
Put x = y = 0 in equation (i),
⇒ f (0) = f(0). f(0)
⇒ f(0).f(0) – f(0) = 0
⇒ f(0) [f(0) – 1] = 0
⇒ f(0) – 1 = 0
∴ f(0) = 1
Put y =0 in equation (i)
⇒ f(0) = f(x).f(0)
\(\Rightarrow f\left( x \right)=\frac{f\left( 0 \right)}{f\left( 0 \right)}\)
∴ f(x) = 1
From question,
\(\Rightarrow \frac{dy}{dx}=f\left( x \right)\)
On substituting value of ‘f(x)’,
\(\Rightarrow \frac{dy}{dx}=1\)
⇒ dy = dx
On integrating both sides,
\(\Rightarrow \int dy=\int dx\)
∴ y = x + C
From question,
⇒ y(0) = 1
⇒ 1 = 0 + C
∴ C = 1
Now,
∴ y = x + 1
From question,
\(\Rightarrow y\left( \frac{1}{4} \right)=\frac{1}{4}+1=\frac{1+4}{4}=\frac{5}{4}\)
\(\Rightarrow y\left( \frac{3}{4} \right)=\frac{3}{4}+1=\frac{3+4}{4}=\frac{7}{4}\)
\(\therefore ~y\left( \frac{1}{4} \right)+y\left( \frac{3}{4} \right)=\frac{5}{4}+\frac{7}{4}=\frac{12}{4}=3\)