Question: find the point on the curve y=x^2 +1 that is closest to the point (18,1). , !

Point is  ======let  be the point on the curve  .using the distance formula, the distance between  and  is    is a point on the curve. therefore,  is  . substitute that in for  so that we only deal with two variables.    we can square both sides so that we work without square roots. when the square of  is at a minimum, then  should also be at a minimum.    if we let  then we can differentiate this to get    . you can factor it using the factor/remainder theorems. if you graph  , there is a zero at  so that is a critical number. (you could have also graphed the original function  with a graphing calculator and find the minimum from there.)to the left of  ,  is negativeto the right of  ,  is positivethis is indeed a minimum.the distance is minimized when  .on the curve, when  ,  so the point is