Let f(x) be a polynomial function such that f(x)+f ( 1 x )=f(x) f ( 1 x ) for all non-zero real x , and f(2)=-7 . The area of the region bounded by the curve y=f(x) and its tangent at the point x=-1 is:

Step-by-step solution

A polynomial satisfying the functional equation f(x)+f ( 1 x )=f(x) f ( 1 x ) is of the form f(x)=1 x^n . Given f(2)=-7 , we have 1 2^n=-7 . Taking the minus sign, 1-2^n=-7 2^n=8 n=3 . Thus, f(x)=1-x^3 . The point of tangency at x=-1 is y=f(-1)=1-(-1)^3=2 . So the point is (-1, 2) . The derivative is f^ (x)=-3 x^2 , so the slope of the tangent at x=-1 is f^ (-1)=-3 . The equation of the tangent line is: y-2=-3(x+1) y=-3 x-1 To find the other point of intersection between the curve and the tangent, solve: 1-x^3=-3 x