f x = x - 14 x + 19 The function f is defined by the given equation. For what

Question

f x = x - 14 x + 19 The function f is defined by the given equation. For what value of x does f x reach its minimum?

Accepted Answer

Choice D is correct. It's given that f x = x - 14 x + 19 , which can be rewritten as f x = x 2 + 5 x - 266 . Since the coefficient of the x 2 -term is positive, the graph of y = f x in the xy -plane opens upward and reaches its minimum value at its vertex. The x -coordinate of the vertex is the value of x such that f x reaches its minimum. For an equation in the form f x = a x 2 + b x + c , where a , b , and c are constants, the x -coordinate of the vertex is - b 2 a . For the equation f x = x 2 + 5 x - 266 , a = 1 , b = 5 , and c = - 266 . It follows that the x -coordinate of the vertex is - 5 2 1 , or - 5 2 . Therefore, f x reaches its minimum when the value of x is - 5 2 . Alternate approach: 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&rsquo;s given that f x = x - 14 x + 19 , it follows that the two x -intercepts of the graph of y = f x in the xy -plane occur when x = 14 and x = - 19 , or at the points 14 , 0 and - 19 , 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 14 , 0 and - 19 , 0 is 14 + - 19 2 , 0 + 0 2 , or - 5 2 , 0 . It follows that f x reaches its minimum when the value of x is - 5 2 . Choice A is incorrect. This is the y -coordinate of the y -intercept of the graph of y = f x in the xy -plane. Choice B is incorrect. This is one of the x -coordinates of the x -intercepts of the graph of y = f x in the xy -plane. Choice C is incorrect and may result from conceptual or calculation errors.

Question f-x-x-