SOLUTION: Given that f(x) = 1 + 4x - x^2 for x ≥ 2. Find the coordinates of the turning point of the function f(x),stating whether it is a maximum or minimum point. Explain why f(x) has an

Algebra ->  Test -> SOLUTION: Given that f(x) = 1 + 4x - x^2 for x ≥ 2. Find the coordinates of the turning point of the function f(x),stating whether it is a maximum or minimum point. Explain why f(x) has an      Log On


   

Question 1184854: Given that f(x) = 1 + 4x - x^2 for x ≥ 2. Find the coordinates of the turning point of the function f(x),stating whether it is a maximum or minimum point. Explain why f(x) has an inverse and find an expression for f^-1(x) in terms of x.
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
%22f%27%28x%29%22+=+4-2x ===> the turning point is at x = 2, since %22f%27%282%29%22+=+0.

%22f%27%27%28x%29%22+=+-2+%3C+0, and in particular %22f%27%27%282%29%22+=+-2+%3C+0 ===> there is a maximum point at x = 2.
(This is to be expected since the coefficient of x%5E2 is negative, and so the parabola opens downward.)

For x%3E=2, f(x) has an inverse function because it is one-to-one over this set. The function is one-to-one because it is strictly decreasing over [2, %2Binfinity).

The inverse function is obtained as follows:
y+=+-x%5E2+%2B+4x+%2B+1 ===> 0+=+-x%5E2+%2B+4x+%2B+%281-y%29 ===> x+=+2+%2B-+sqrt%285-y%29

after solving x in terms of y using the quadratic formula. Choose x+=+2+%2B+sqrt%285-y%29 since x%3E=2.

Interchanging the places of x and y, the inverse function is therefore f%5E%28-1%29%28x%29+=+2+%2B+sqrt%285-x%29.