The function f is defined as f(x)=x^(2)-1. What is the x-coordinate of the point on the function's graph that is closest to the origin? Choose all answers that apply: A -(sqrt3)/(3) B -(sqrt2)/(2) c 0 D (sqrt3)/(3) E (sqrt2)/(2)

Question

The function  f is defined as  f(x)=x^(2)-1. What is the  x-coordinate of the point on the function's graph that is closest to the origin? Choose all answers that apply: A  -(sqrt3)/(3) B  -(sqrt2)/(2) c 0 D  (sqrt3)/(3) E  (sqrt2)/(2)

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Define Function: The function is f(x) = x^2 - 1. To find the point closest to the origin, we need to minimize the distance from the point (x, f(x)) to the origin (0,0).Calculate Distance Formula: The distance from a point (x, y) to the origin is given by the formula D = sqrt(x^2 + y^2). For the function f(x), this becomes D = sqrt(x^2 + (x^2 - 1)^2).Minimize Distance Formula: To minimize D, we minimize D^2 to avoid dealing with the square root. So we minimize x^2 + (x^2 - 1)^2.Expand and Simplify: Expanding D^2 gives us x^2 + x^4 - 2x^2 + 1. Simplifying, we get D^2 = x^4 - x^2 + 1.Find Derivative: To find the minimum, we take the derivative of D^2 with respect to x and set it equal to zero. So, d(D^2)/dx = 4x^3 - 2x.Set Derivative Equal to Zero: Setting the derivative equal to zero gives us 4x^3 - 2x = 0. Factoring out 2x, we get 2x(2x^2 - 1) = 0.Solve for Solutions: Setting each factor equal to zero gives us two solutions: x = 0 and 2x^2 - 1 = 0. Solving the second equation gives x = ±sqrt(1/2).

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