a. Given that y = f(x) and f(0) = -2, by the fundamental theorem of calculus we have
[tex]\displaystyle \frac{dy}{dx} = 3x^2 + 4x + k \implies y = f(0) + \int_0^x (3t^2+4t+k) \, dt[/tex]
Evaluate the integral to solve for y :
[tex]\displaystyle y = -2 + \int_0^x (3t^2+4t+k) \, dt[/tex]
[tex]\displaystyle y = -2 + (t^3+2t^2+kt)\bigg|_0^x[/tex]
[tex]\displaystyle y = x^3+2x^2+kx - 2[/tex]
Use the other known value, f(2) = 18, to solve for k :
[tex]18 = 2^3 + 2\times2^2+2k - 2 \implies \boxed{k = 2}[/tex]
Then the curve C has equation
[tex]\boxed{y = x^3 + 2x^2 + 2x - 2}[/tex]
b. Any tangent to the curve C at a point (a, f(a)) has slope equal to the derivative of y at that point:
[tex]\dfrac{dy}{dx}\bigg|_{x=a} = 3a^2 + 4a + 2[/tex]
The slope of the given tangent line [tex]y=x-2[/tex] is 1. Solve for a :
[tex]3a^2 + 4a + 2 = 1 \implies 3a^2 + 4a + 1 = (3a+1)(a+1)=0 \implies a = -\dfrac13 \text{ or }a = -1[/tex]
so we know there exists a tangent to C with slope 1. When x = -1/3, we have y = f(-1/3) = -67/27; when x = -1, we have y = f(-1) = -3. This means the tangent line must meet C at either (-1/3, -67/27) or (-1, -3).
Decide which of these points is correct:
[tex]x - 2 = x^3 + 2x^2 + 2x - 2 \implies x^3 + 2x^2 + x = x(x+1)^2=0 \implies x=0 \text{ or } x = -1[/tex]
So, the point of contact between the tangent line and C is (-1, -3).
