find y as a function of x given that d2y/dx2=6x-4 and that when y=4, x=1 and y=2, x=-1

Question

the answer given is y=x^3-2x^2+5

Answer 1

y''=6x-4; y'=3x^2-4x+a; y=x^3-2x^2+ax+b, where a and b are constants. We find a and b by using the x-y information given.

When y=4, x=1, so 4=1-2+a+b; a+b=5; when y=2, x=-1, so 2=-1-2-a+b; -a+b=5; therefore b=5 and a=0.

So y=x^3-2x^2+5.