Let f:[1,∞)→[2,∞) be a differentiable function. If ∫1xf(t)dt=5xf(x)−x5−9 for all x≥1, then the value of f(3) is:
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Step-by-Step Solution
Step 1: Differentiate both sides
To solve this problem, we need to find the function f(x). The given equation involves an integral, so we differentiate both sides with respect to x. This will eliminate the integral on the left side using the Fundamental Theorem of Calculus.
Step 2: Apply Fundamental Theorem of Calculus and Product Rule
Applying the Fundamental Theorem of Calculus, the derivative of the left side is f(x). On the right side, we use the product rule for 5x f(x) and the power rule for x5. The derivative of the constant 9 is 0.
Step 3: Rearrange the equation into a linear first-order differential equation
Now, we rearrange the equation to group terms involving f(x) and f'(x). This results in a first-order linear differential equation.
Step 4: Solve the differential equation
We can solve this differential equation by recognizing it as an exact derivative. Dividing by x and multiplying by x3 (which is an integrating factor), we get 5x4f′(x)+4x3f(x)=5x7, which simplifies to dxd(x4f(x))=x7. Integrating both sides with respect to x gives x4f(x)=8x8+C.
Step 5: Find the constant of integration C
To find the constant C, we use the original integral equation. Substituting x=1 into the original equation gives ∫11f(t)dt=5(1)f(1)−15−9. The left side is 0. This simplifies to 0=5f(1)−10, so f(1)=2. Now, substitute x=1 and f(1)=2 into x4f(x)=8x8+C to find C. This gives 14(2)=818+C, so 2=81+C, which means C=815.
Step 6: Determine f(x)
With C=815, the function becomes x4f(x)=8x8+815. Dividing by x4, we get the explicit form of f(x).
Step 7: Calculate f(3)
Finally, substitute x=3 into the expression for f(x) to find the value of f(3).