find a and b so that f(x)=x^2+ax+b has a local min at (6,-5)

Question

i found the derivative:
f'=2x+a

then i found a

0=2x+a
0=2(6)+a
0=12+a
a=-12

but i can't seem to get the correct value for b. i plugged -5 into the original function, but my book says the answer is b=31. how do i get 31??

Answer 1

Here's how you get 31:

You are correct that a = -12.
Do, now the function becomes:

f(x) = x^2 - 12x + b.

Now, we know that x is 6 and y is -5. So plug these in:

-5 = 6^2 - 12(6) + b

solve for b and get 31.

Do you understand?

Answer 2

yes, i feel like such an idiot, but it makes sense now. thank you!!!