f x = x - 2 x + 15 The function f is defined by the given equation. For what

Question

f x = x - 2 x + 15 The function f is defined by the given equation. For what value of x does f x reach its minimum?

Accepted Answer

The correct answer is - 13 2 . The value of x for which f x reaches its minimum can be found by rewriting the given equation in the form f x = x - h 2 + k , where f x reaches its minimum, k , when the value of x is h . The given equation, f x = x - 2 x + 15 , can be rewritten as f x = x 2 + 13 x - 30 . By completing the square, this equation can be rewritten as f x = x 2 + 13 x + 13 2 2 - 30 - 13 2 2 , which is equivalent to f x = x + 13 2 2 - 289 4 , or f x = x - - 13 2 2 - 289 4 . Therefore, f x reaches its minimum when the value of x is - 13 2 . Note that -13/2 and -6.5 are examples of ways to enter a correct answer. Alternate approach: The graph of y = f x in the xy -plane is a parabola. The value of x for the vertex of a parabola is the x -value of the midpoint between the two x -intercepts of the parabola. Since it's given that f x = x - 2 x + 15 , it follows that the two x -intercepts of the graph of y = f x in the xy -plane occur when x = 2 and x = - 15 , or at the points 2 , 0 and - 15 , 0 . The midpoint between two points, x 1 , y 1 and x 2 , y 2 , is x 1 + x 2 2 , y 1 + y 2 2 . Therefore, the midpoint between 2 , 0 and - 15 , 0 is 2 - 15 2 , 0 + 0 2 , or - 13 2 , 0 . It follows that f x reaches its minimum when the value of x is - 13 2 . Note that -13/2 and -6.5 are examples of ways to enter a correct answer.

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