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Find, correct to two decimal places, the coordinates of the point on the curve $ y = \sin x $ that is closest to the point $ (4, 2) $.

Point closest to $(4,2)$ is $(2.65,0.47)$

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left? Yeah. Adams. We're asked to find correct two decimal places. The coordinates of the point on the curve y equals sine of X that is closest to the 0.42 Now to do this first, let's find the distance between the curve and this point. This distance D of X y is the square root of X minus four squared plus why minus two squared. And so as a function of X d F X is the square root of X minus four squared plus sine of X minus two squared Now our distance function D thank yeah is yeah, I love non negative and is increasing. Therefore, Stubborn said, I'm a butterfly. Therefore, it follows that our distance D is minimized when it's square she'll call F, which is D squared is minimized. Fairfield Comic seven pieces Function F of X. This is D squared of X, which is X minus four squared, plus the sine of X minus two squared his hands now to find the court. It's the point closest to 42 I'm going to take the derivative at prime of X, so this is two times X minus four plus two times sine X minus two times that are the inside, which is co sign of X and then to find critical values that set the sequel to zero. She has you cheer so we actually end up with a trigger the metric equation, which is not really easy to solve. However, if you do solve it, Yes, Yeah, and yeah, that's it comes to Alias Sydney, the 23rd, Melvin 20 Brisbane 28. It will get a few different values for X the value of X. Yeah, bye, Everybody, That's closest is X equals about 2.65 This is the most likely solution. Let's test this by finding the second derivative. After all, prime of X. This is going to be too plus and then two times three of them of sign X cosign of x times the derivative or then we have co sign of X by product rule. MM plus two times sine of X minus two times negative sine of X. And so if we plug in 2.65 we get two plus two times the co sign of 2.65 squared minus two times the sine squared of 2.65 plus mhm to its Yeah, four times the sine of 265 Yes, right. Yeah. And in fact, if you plug this into your calculator and evaluate, you'll find this is greater than zero. So we find by the second derivative test that our function F as a relative maximum value at X is approximately 2.65 And okay, if we compare two other exes that were answers to F prime of X equals zero You see that F has in fact, an absolute yes. So what? Mhm this ball. I'm sorry. Not Max. I mean relative minimum, because the second derivative is greater than zero. So it is an absolute minimum intuitively, at about X equals 2.65 be so easy to find the Why coordinate Well, why is equal approximately to the sign of 265 which is about 0.47 and so correct. The two decimal places the coordinates of the point on the curve like will Synnex that is closest to the 20.0.42 are 2.65 0.47