Question

If f(x)=ax^2+b, f(−1)=−2 and f′(−1)=8, find the coefficients a and b.
Answers:
a=

b=

Answer 1

@Michele_Laino

Answer 2

@dan815

Answer 3

by substitution, we get:
\[f\left( { - 1} \right) = a{\left( { - 1} \right)^2} + b = a + b = - 2\]

Answer 4

okay

Answer 5

now, the first derivative of \(f(x)\), is:
\[f'\left( x \right) = 2ax\]
please replace \(x=-1\), what do you get?

Answer 6

-2a

Answer 7

so the coefficient is -2 for a

Answer 8

so, we can write:
\[ - 2a = 8\]

Answer 9

please solve for \(a\)

Answer 10

a=-4

Answer 11

ok!
so, substituting such value for \(a\) into the first equation, we can write:
\[\begin{gathered}
a + b = - 2 \hfill \\
\hfill \\
- 4 + b = - 2 \hfill \\
\end{gathered} \]
please solve for \(b\)

Answer 12

2

Answer 13

that's right!

Answer 14

that makes a lot more sense now :D

Answer 15

:)