If y= ax-b / (x-1) (x-4) has a turning point at A(2,-1), find the values of a and b so - Maths - Application of Derivatives

Question

If y= ax-b / (x-1) (x-4) has a turning point at A(2,-1), find the
values of a and b so that y has maxima at the point A

Neha Sethi · Accepted Answer

Dear student
We have, y=ax-bx-1x-4=ax-bx2-5x+4   i⇒dydx=x2-5x+4a-ax-b2x-5x2-5x+4  ii⇒dydxp=4-10+4a-2a-b4-54-10+42=-b4Since P is a turning point of the curve i,So,dydxp=0⇒-b4=0⇒b=0  iiiSince P2,-1 lies on y=ax-bx-1x-4
So, -1=2a-b2-12-4⇒-1=2a-b-2⇒2a-b=2From  iii and iv, we get a=1 and b=0Substituting the values of a and b in ii we getdydx=x2-5x+4-x2x-5x2-5x+42=-x2+4x2-5x+42⇒d2ydx2=x2-5x+42-2x--x2+42x2-5x+42x-5x2-5x+42Now, dydx2,-1=0 and d2ydx22,-1=-2-4-23=-1<0So, y is maximum  at P when a=1 and b=0
Regards

If y= ax-b / (x-1) (x-4) has a turning point at A(2,-1), find the values of a and b so that y has maxima at the point A.