the curve y= (x2+ax+ b) / x-10 has a turning point at (4, 1) fine the values of a and b and show that y is maximum at this point

We are giveny=x2+ax+bx-10dydx=x-102x+a-x2+ax+bx-102=2x2+a-20x-10a-x2+ax+bx-102dydx=x2-20x-10a-bx-102As, the curve has a turning point at 4,1.  Therefore,dydx4,1=042-204-10a-b4-102=016-80-10a-b=010a+b=-66                   ..........iAlso, the point 4,1 lies on the curve, Therefore1=42+4a+b4-104a+b+16=-64a+b=-22                     ........iiSubracting ii from i, we get6a=-44 a=-223.Putting a=-223 in equation ii,we get-883+b=-22b=223.Now, d2ydx2=2x-103-2x-10x2-20x-10a-bx-104=2x-10-2x2-20x-10a-bx-103d2ydx24,1=2-6-216-80-10a-b-63=-13-2-66+10223-223-216                                    =-13-2-66+66-216=-13<0As, d2ydx24,1<0  at4,1. The given curve has apoint of maximum at this point.

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