Question

Find constants a and b so that the minimum for the parabola f(x)=x^2+ax+b is at the point (5,7)

Answer 1

To find the minimum of the parabola, the equation must be derived and equated to zero as follows.

F(x)=
`x^(2)` + ax + b

F'(x) = 2x + a = 0

The minumum is at point (5,7). Thus, we substitute x with 5 to find a.

2(5) + a = 0

a = -10

To find for b, we replace x, y, and a values to the original equation of the parabola as follows.

F(x)=
`x^(2)` + ax + b

y=
`x^(2)` + ax + b

7 =
`5^(2)` + (-10)(5) + b

b = 32

Therefore the answer is: a = -10 and b = 32