HW22 Optımization: Proolem 2 (1 point) Find the point on the line $2 x+1 y-2=0$ which is closest to the point $(-5,-1)$. Answer: Note: You can earn partial credit on this problem. Preview My Answers Submit Answers You have attempted this problem 0 times. You have 10 attempts remaining. Email Instructor

Step 1 ：Let the point on the line \(2x + y - 2 = 0\) be \((x, y)\)

Step 2 ：Substitute \(y\) from the line equation to get \(y = -2x + 2\)

Step 3 ：Use the distance formula squared to get the function to minimize: \((x - (-5))^2 + (-2x + 2 - (-1))^2\)

Step 4 ：Simplify the function to get: \((x + 5)^2 + (3 - 2x)^2\)

Step 5 ：Take the derivative of the function with respect to \(x\): \(10x - 2\)

Step 6 ：Set the derivative equal to zero and solve for \(x\): \(10x - 2 = 0\), \(x = \frac{1}{5}\)

Step 7 ：Substitute \(x\) back into the line equation to find \(y\): \(y = -2(\frac{1}{5}) + 2 = \frac{8}{5}\)

Step 8 ：\(\boxed{\left(\frac{1}{5}, \frac{8}{5}\right)}\) is the point on the line closest to \((-5, -1)\)