Question

Given \(\mathrm{f}(\mathrm{x})\) is a differentiable function of \(\mathrm{x}\), satisfying \(f(x) . f(y)=f(x)+f(y)+f(x y)-2\) and that \(f(2)=5\). Then \(\mathrm{f}(3)\) is equal to (A) 10 (B) 24 (C) 15 (D) none

Short answer

Based on the given function properties and the step-by-step solution, the value of f(3) is found to be 3. However, this value doesn't match with any of the given choices. Thus, the correct answer is (D) none.

Step-by-step solution

Prove that the function f(x) is uniquely determined by its properties

Here we need to show that there is only one function that satisfies the given condition. Let's set \(x=y=1\): $$f(1) . f(1) = f(1) + f(1) + f(1) - 2$$ $$f(1)^2 - 2f(1) + 1 = 0$$ $$(f(1) - 1)^2 = 0$$ Therefore, \(f(1) = 1\). Now, let's set \(y = 1\): $$f(x)f(1) = f(x) + f(1) + f(x) - 2$$ $$f(x) = f(x) + 1 + f(x) - 2$$ This simplifies to \(f(x) = 1\). This shows that the function has the form f(x) = x.

Find an expression for the derivative of f(x)

Since we have found that \(f(x) = x\), we can now differentiate f(x) with respect to x: $$f'(x) = \frac{d}{dx} f(x) = \frac{d}{dx} x = 1 $$

Utilize the given information to find f(3)

Since f(x) = x, we can simply find f(3) as follows: $$f(3) = 3$$ However, this value doesn't match with any of the given choices. Hence, the correct answer is (D) none.