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Find the point on the curve $ y = \sqrt{x} $ that is closest to the point $ (3, 0) $.

$P\left(\frac{5}{2}, \frac{\sqrt{10}}{2}\right)$

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whereas to find the point on the curve Y equals the square root of X, it is closest to the 0.30 being able to be humiliated. Well, what is the distance from this curve to this point? This distance, which are called D as a function of X and Y. It's the square root of X minus three squared plus y minus zero squared, which is the square root of X minus three squared plus y squared. This is the same as just X, I think part every assaulted. Now we know that our distance D is going to be minimized when function F, which is D squared, is minimized as well. This is because D is a increasing positive function, so F as a function of X. This is D as a function of X squared, which is X minus three squared plus X now to find the minimum of effort going to take the derivative F prime of X, just two times X minus three plus one, find critical values to set this equal to zero. Yeah, so we have X minus three equals negative. One half, or X equals negative one half plus three, which is, I'm still positive. Five hats. Yeah, this one. Actually, this was now the second derivative F double prime of X. This is equal to to which is, of course, always positive. Therefore, it follows that our function f has an absolute minimum at its critical value. Well, it has an absolute minimum on its domain, from zero to infinity. Well, actually notes it's on all and he got his son who is living and breathing on all real numbers X. And it has it at the critical value. X equals five halves. Yeah, inside yourself. So the point that's closest has coordinates X equals five halves. And why is the square root of five halves There's no Freddy hurt? Yes, it's which, of course, is the same as the square root of 10 over to. And so we have the points five halves. Route 10 over to this is the point closest to the 0.0.30 on the curve. White will swear to vex, so mm