JEE Mains · Maths · STD 12 - 9. differential equations
Let \(f: [1, \infty) \rightarrow \mathbf{R}\) be a differentiable function defined as \(f(x) = \int_1^x f(t)\,dt + (1-x)(\log_e x - 1) + e\). Then the value of \(f(f(1))\) is :
- A \((1 + e^e)\)
- B \((1 + e)\)
- C \((1 + e + e^e)\)
- D \(1 + 2e\)
Answer & Solution
Correct Answer
(A) \((1 + e^e)\)
Step-by-step Solution
Detailed explanation
Given \(f(x) = \int_1^x f(t)\,dt + (1-x)(\log_e x - 1) + e\) Substituting \(x = 1\) in the given equation: \(f(1) = \int_1^1 f(t)\,dt + (1-1)(\log_e 1 - 1) + e\) \(f(1) = 0 + 0 + e = e\) Differentiating the given equation with respect to \(x\) using the Leibniz rule:…
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