Step by step solution to "Find the points on the graph of the..."

cristinamahumas

Answered question

2022-01-14

Find the points on the graph of the function that are closest to the given point y=x^2, (0,3)

Vasquez

Expert2022-02-06Added 669 answers

Let the point is $(x, y)$

Distance, $d^{2} = x^{2} + (y - 3)^{2}$

$d^{2} = x^{2} + (x^{2} - 3)^{2}$

$f (x) = x^{2} + (x^{2} - 3)^{2}$

$f^{'} (x) = 2 x + 4 x (x^{2} - 3)$

$f^{'} (x) = 0$

$2 x + 4 x (x^{2} - 3) = 0$

$x = 0, x = \sqrt{\frac{5}{2}}, x = - \sqrt{\frac{5}{2}}$

$y = x^{2}$

so $y = 0, y = \frac{5}{2}, y = \frac{5}{2}$

Points are $(- \sqrt{\frac{5}{2}}, \frac{5}{2}), (0, 0), (\sqrt{\frac{5}{2}}, \frac{5}{2})$

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