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# minimum

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What is the smallest distance between the origin and a point on the graph of y = x^2 - 3?

Jun 24, 2021

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The distance from a point on the graph to the origin can be expressed as, in terms of x, as $$\sqrt{x^2+(x^2-3)^2}=\sqrt{x^4-6x^2+9+x^2}=\sqrt{x^4-5x^2+9}$$

the quartic inside the radical is just a quadratic in terms of x^2, so we can rewrite it in vertex form:

$$\sqrt{x^4-5x^2+9}=\sqrt{x^4-5x^2+(\frac{5}{2})^2+9-(\frac{5}{2})^2}=\sqrt{(x^2-\frac{5}{2})^2+\frac{11}{4}}$$

since $$(x^2-2.5)^2$$will always be nonnegative for real numbers, the minimum value for $$\sqrt{(x^2-2.5)^2+\frac{11}{4}}$$ is just equal to $$\boxed{\frac{\sqrt{11}}{2}}$$ (specifically when x is equal to $$\pm\sqrt{2.5}$$)

P.S. my method seems like total overkill for this problem so if anyone has an easier solution, please post it

Jun 25, 2021