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------------------------------------------------ If \( f(0) = 1 \), \( f(2) = 3 \), \( f'(0) = 5 \), and \( f'(2) = 7 \), find the value of

\[
\int_{0}^{2} 3x \cdot f''(x) \, dx
\]

A. 12
B. -12
C. 24
D. -24

Answer :

Final answer:

To find the value of the integral, use the Second Fundamental Theorem of Calculus and the given values to calculate the integral.

Explanation:

To find the value of the integral, we need to use the Second Fundamental Theorem of Calculus, which states that if we have a continuous function f(x) and its derivative f'(x), then the integral of f'(x) from a to b is equal to f(b) - f(a).

In this case, we are given f(0) = 1, f(2) = 3, f'(0) = 5, and f'(2) = 7. We need to find the integral of 3x * f''(x) from 0 to 2.

Using the Second Fundamental Theorem of Calculus, we have:

∫ 2 to 0 (3x * f''(x)) dx = [3x * f'(x)]02 = (3 * 2 * f'(2)) - (3 * 0 * f'(0)) = 6 * 7 - 0 = 42.

Therefore, the value of the integral is 42.

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