Mathematics

Mathematics

Problem

Let f(x)f(x) be a function which satisfy the equation f(xy)=f(x)+f(y)f(xy) = f(x) + f(y) for all x>0,y>0x > 0, y > 0 such that f′(1)=2f'(1) = 2. Find the area of the region bounded by the curves y=f(x),y=∣x3−6x2+11x−6∣y = f(x), y = |x^3 - 6x^2 + 11x - 6| and x=0x = 0.
Intuitive explanation

The functional equation f(xy)=f(x)+f(y)f(xy) = f(x) + f(y) is Cauchy's logarithmic form—it uniquely determines ff as f(x)=cln⁡xf(x) = c\ln x for some constant cc. Use the derivative condition f′(1)=2f'(1) = 2 to find cc, then identify where the absolute value expression changes sign to set up the correct integral bounds and determine which curve is on top.

Intuitive explanation

The functional equation f(xy)=f(x)+f(y)f(xy) = f(x) + f(y) is Cauchy's logarithmic form—it uniquely determines ff as f(x)=cln⁡xf(x) = c\ln x for some constant cc. Use the derivative condition f′(1)=2f'(1) = 2 to find cc, then identify where the absolute value expression changes sign to set up the correct integral bounds and determine which curve is on top.

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