Find the expression for function which has a 2nd derivative of f"(x) = 6x + 6 which passes through point (1, -2) with a slope value of 2 at that...

The function to be found has a second derivative f''(x) =
6x + 6

Integrating
f''(x),

f'(x) = Int[ f''(x) dx] = Int[6x + 6
dx]

=> 3x^2 + 6x +
C1

f(x) = Int[ f'(x) dx] = Int [ 3x^2 + 6x + C
dx]

=> x^3 + 3x^2 + C1*x +
C2

As the function has a slope of 2 at (1,
-2)

3*1^2 + 6*1 + C1*x =
-2

=> 3 + 6 + C1*x =
-2

=> 9 + C1*1 =
-2

=> C1 = -11

As the
function passes through (1 , -2)

1^3 + 3*1^2 - 11*1 + C2 =
-2

=> 1 + 3 - 11 + C2 =
-2

=> 4 - 7 + C2 =
-2

=> C2 =
5

The required function is f(x) = x^3 + 3x^2
- 11x + 5