Final Answer:
The value of ∫x⋅f ′′ (x)dx is 24.
Step-by-step explanation:
Given the function values f(6) = -14, f'(6) = 12, f(9) = -17, and f'(9) = -9. To find ∫x⋅f ′′ (x)dx, we'll need to employ the Fundamental Theorem of Calculus and integrate by parts.
Firstly, using the second derivative, f''(x), we can find the value of f''(x) at x = 6 and x = 9. By calculating the differences between f'(6) and f'(9) as -9 - 12 = -21, and the differences between f(9) and f(6) as -17 - (-14) = -3, we get the value of ∫f ′′ (x)dx as -3.
Now, to find ∫x⋅f ′′ (x)dx, integrate the function x with respect to x, resulting in (x^2)/2, and multiply this by the value of f''(x). Thus, ∫x⋅f ′′ (x)dx = (x^2)/2 * (-3). Next, evaluate this expression from x = 6 to x = 9, which yields ((9^2)/2 - (6^2)/2) * (-3) = (81/2 - 36/2) * (-3) = (45/2) * (-3) = -135/2 = -67.5.
Therefore, the value of ∫x⋅f ′′ (x)dx is -67.5. However, considering the integral is the area under the curve, and as integration can't result in a negative area, it's crucial to take the absolute value, resulting in ∫x⋅f ′′ (x)dx = |-67.5| = 67.5.