Question

For the function f(x)=ax^3+bx^2-5x+9, determine the values of a and b so that f(-1)=12 and f'(-1)=3

Answer 1

If

f(x) = ax ³ + bx ² - 5x + 9

then

f '(x) = 3ax ² + 2bx - 5

Given that f (-1) = 12 and f '(-1) = 3, we get the system of equations

-a + b + 5 + 9 = 12

3a - 2b - 5 = 3

or

-a + b = -2

3a - 2b = 8

Multiply through the first equation by 2 and add it to the second one to eliminate b and solve for a :

2(-a + b) + (3a - 2b) = 2(-2) + 8

-2a + 2b + 3a - 2b = -4 + 8

a = 4

Substitute this into the first equation above to solve for b :

-4 + b = -2

b = 2