To find the value of [tex]\(\int_1^4 x f^{\prime \prime}(x) \, dx\)[/tex], let's proceed step-by-step.
1. Given Data:
- [tex]\(f(1) = 2\)[/tex]
- [tex]\(f(4) = 6\)[/tex]
- [tex]\(f'(1) = 3\)[/tex]
- [tex]\(f'(4) = 2\)[/tex]
- [tex]\(f''(x)\)[/tex] is continuous
2. Integration by Parts:
We use integration by parts for the integral [tex]\(\int_1^4 x f^{\prime \prime}(x) \, dx\)[/tex]. Let:
- [tex]\(u = x\)[/tex]
- [tex]\(dv = f''(x) dx\)[/tex]
Then, we compute:
- [tex]\(du = dx\)[/tex]
- [tex]\(v = f'(x)\)[/tex] (because the integral of [tex]\(f''(x)\)[/tex] is [tex]\(f'(x)\)[/tex])
According to integration by parts,
[tex]\[
\int u \, dv = uv - \int v \, du.
\][/tex]
Substituting in our functions, we get:
[tex]\[
\int_1^4 x f^{\prime \prime}(x) \, dx = \left[ x f'(x) \right]_1^4 - \int_1^4 f'(x) \, dx.
\][/tex]
3. Evaluate [tex]\( \left[ x f'(x) \right]_1^4\)[/tex]:
[tex]\[
[x f'(x)]_1^4 = 4f'(4) - 1f'(1).
\][/tex]
Plugging in the given values:
[tex]\[
4f'(4) - 1f'(1) = 4 \cdot 2 - 1 \cdot 3 = 8 - 3 = 5.
\][/tex]
4. Evaluate [tex]\(\int_1^4 f'(x) \, dx\)[/tex]:
- By the Fundamental Theorem of Calculus, [tex]\(\int_1^4 f'(x) \, dx = f(4) - f(1)\)[/tex].
- Plugging in the given values:
[tex]\[
f(4) - f(1) = 6 - 2 = 4.
\][/tex]
5. Combine Results:
Substitute these results back into the expression from step 2:
[tex]\[
\int_1^4 x f^{\prime \prime}(x) \, dx = 5 - 4 = 1.
\][/tex]
Thus, the value of [tex]\(\int_1^4 x f^{\prime \prime}(x) \, dx\)[/tex] is [tex]\(\boxed{1}\)[/tex].
