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) $-\frac{\sqrt{3}}{3}$

B $-\frac{\sqrt{2}}{2}$
c] 0
ㅁ $\frac{\sqrt{3}}{3}$
E $\frac{\sqrt{2}}{2}$

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) $-\frac{\sqrt{3}}{3}$

B $-\frac{\sqrt{2}}{2}$
c] 0
ㅁ $\frac{\sqrt{3}}{3}$
E $\frac{\sqrt{2}}{2}$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the $x$-coordinate of the point on the graph of $f(x)=x^{2}-1$ that is closest to the origin, we need to minimize the distance from the point $(x, f(x))$ to the origin $(0,0)$. ‖ ‖ step 2 ⋮ The distance squared from $(x, f(x))$ to the origin is given by $D(x) = x^2 + (x^2 - 1)^2$. We minimize $D(x)$ to find the closest point. ‖ ‖ step 3 ⋮ According to the asksia-ll calculation list, the minimum value of $(x^2 - 1)^2 + x^2$ occurs at $x = -\frac{1}{\sqrt{2}}$ and $x = \frac{1}{\sqrt{2}}$. ‖ ∻Answer∻ ⚹ B $-\frac{\sqrt{2}}{2}$ and E $\frac{\sqrt{2}}{2}$ ⚹ ∻Key Concept∻ ⚹Minimizing Distance to the Origin⚹ ∻Explanation∻ ⚹The $x$-coordinates that minimize the distance from the graph of $f(x)$ to the origin are the values of $x$ that minimize the function $D(x) = x^2 + (x^2 - 1)^2$.⚹◊What method can be used to find the minimum of a function defined by $f(x)=x^{2}-1$ closest to the origin?⍭ Generate me a similar question◊

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