A curve satisfies d²y/dx² = 6x - 4. At the point (1, 2) the slope of the curve is 3…

Step 1. First integration. $\dfrac{dy}{dx} = 3x^2 - 4x + C_1$. At $x = 1$ the slope is $3$, so $3 - 4 + C_1 = 3$ and $C_1 = 4$.

Step 2. Second integration. $y = x^3 - 2x^2 + 4x + C_2$. At $(1, 2)$ we get $1 - 2 + 4 + C_2 = 2$, so $C_2 = -1$.

Step 3. Evaluate. The slope at $x = 2$ is $12 - 8 + 4 = 8$. The height is $y = 8 - 8 + 8 - 1 = 7$.