Question

Find constants a and b so that the minimum for the parabola f left-parenthesis x right-parenthesis equals x squared plus a x plus b is at the point left-parenthesis 6 comma 7 right-parenthesis. [hint: begin by finding the critical point in terms of

a.]

Answer 1

f left-parenthesis x right-parenthesis equals x squared plus a x plus b is at the point left-parenthesis 6 comma 7 right-parenthesis has a vertex at -b/2a

Answer 2

`a=-12` and
`b=43`

Explanation:

To find the constants, we need to replace the given point
`(6;7)` into the given quadratic function:
`f(x)=x^(2)+ax+b`

So, from the problem we have
`x=6` and
`y=7`. Then:

`f(x)=x^(2)+ax+b\\7=(6)^(2)+a(6)+b\\7=36+6a+b\\7-36=6a+b\\-29=6a+b`

However, if we have a minimum at
`(6;7)`, the first derivate of the function must be zero at 6.

`f'(x)=2x+a=\\0=2(6)+a\\-12=a`

Now, we substitute this value in the previous expression:

`-29=6a+b\\-29=6(-12)+b\\-29+72=b\\b=43`

Therefore, the values of the constants are
`a=-12` and
`b=43`