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

Full solution

Q. The function

f

is defined as

f(x)=x^{2}-1

\newline

What is the

x

-coordinate of the point on the function's graph that is closest to the origin?

\newline

Choose all answers that apply:

\newline

-\frac{\sqrt{3}}{3}

\newline

-\frac{\sqrt{2}}{2}

\newline

0

\newline

\frac{\sqrt{3}}{3}

\newline

\frac{\sqrt{2}}{2}

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, $\frac{d(D^2)}{dx} = 4x^3 - 2x$ .
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 $x = \pm\sqrt{1/2}$ .