Determine equations for the tangents...

Question

Determine the values of a and b for the function f(x) = ax^3 + bx^2 +3x -2 given that f(2) = 10 and f'(-1) = 14

Answer 1

Determine the values of a and b for the function f(x) = ax^3 + bx^2 +3x -2 given that f(2) = 10 and f '(-1) = 14.

 

f '(x) = 3ax^2 + 2bx + 3

f '(-1) = 14 = 3a + 2b + 3, 3a + 2b = 11

 

f(2) = 10 = a(2)^3 + b(2)^2 + 3(2) - 2

10 = 8a + 4b + 4

5 = 4a + 2b + 2

 

4a + 2b = 3

-3a - 2b = -11

 

a = -8

 

4(-8) + 2b = 3

2b = 3 + 32 = 35

 

b = 35/2

 

f(x) = -8x^3 + (35/2)x^2 + 3x - 2

Answer 2

f(x)=ax^3+bx^2+3x-2

f(2)=a(2^3)+b(2^2)+3(2)-2=10

8a+4b+6-2=10

8a+4b+4=10

8a+4b=6

 

f(-1)=a((-1)^3)+b((-1)^2)+3(-1)-2=14

-a+b-3-2=14

-a+b-5=14

-a+b=19

b=a+19

substitute into 8a+4b=6

8a+4(a+19)=6

8a+4a+76=6

12a=-70

a=-70/12

a=-35/6=-5  5/6

b=(-35/6)+19

b=13 1/6=79/6