To determine the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] for the function [tex]\(f(x) = ax^3 + bx^2 - 5x + 9\)[/tex] such that [tex]\(f(-1) = 12\)[/tex] and [tex]\(f'(-1) = 3\)[/tex], follow these steps:
1. Substitute [tex]\(x = -1\)[/tex] into [tex]\(f(x)\)[/tex] and set it equal to 12:
[tex]\[
f(-1) = a(-1)^3 + b(-1)^2 - 5(-1) + 9 = 12
\][/tex]
Simplify this equation:
[tex]\[
-a + b + 5 + 9 = 12 \implies -a + b + 14 = 12
\][/tex]
Rearrange this to isolate [tex]\(b\)[/tex]:
[tex]\[
-a + b = -2 \implies b = a - 2 \quad \text{(Equation 1)}
\][/tex]
2. Find the derivative [tex]\(f'(x)\)[/tex]:
[tex]\[
f'(x) = \frac{d}{dx}(ax^3 + bx^2 - 5x + 9) = 3ax^2 + 2bx - 5
\][/tex]
3. Substitute [tex]\(x = -1\)[/tex] into [tex]\(f'(x)\)[/tex] and set it equal to 3:
[tex]\[
f'(-1) = 3a(-1)^2 + 2b(-1) - 5 = 3
\][/tex]
Simplify this equation:
[tex]\[
3a - 2b - 5 = 3
\][/tex]
Rearrange this to isolate [tex]\(2b\)[/tex]:
[tex]\[
3a - 2b - 5 = 3 \implies 3a - 2b = 8 \quad \text{(Equation 2)}
\][/tex]
4. Substitute Equation 1 (b = a - 2) into Equation 2:
[tex]\[
3a - 2(a - 2) = 8
\][/tex]
Simplify and solve for [tex]\(a\)[/tex]:
[tex]\[
3a - 2a + 4 = 8 \implies a + 4 = 8 \implies a = 4
\][/tex]
5. Use the value of [tex]\(a\)[/tex] to find [tex]\(b\)[/tex] using Equation 1 (b = a - 2):
[tex]\[
b = 4 - 2 = 2
\][/tex]
Therefore, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that satisfy the conditions [tex]\(f(-1) = 12\)[/tex] and [tex]\(f'(-1) = 3\)[/tex] are:
[tex]\[
a = 4 \quad \text{and} \quad b = 2
\][/tex]
