Let k 0 be a constant. A differentiable function f: [0, 2 ] R ^+ satisfies f(x) + ₀^ x f(t) k - ( _e f(t))^2 dt = e^ k for all x [0, 2 ] . If f ( 3 ) = e^2 , then the value of k is equal to

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Let k > 0 be a constant. A differentiable function f: [0, 2 ] R ^+ satisfies f(x) + ₀^ x f(t) k - ( _e f(t))^2 dt = e^ k for all x [0, 2 ] . If f ( 3 ) = e^2 , then the value of k is equal to

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Correct answer: 16

Given the integral equation: f(x) + ₀^ x f(t) k - ( _e f(t))^2 dt = e^ k Putting x = 0 , we get: f(0) + 0 = e^ k f(0) = e^ k Differentiating the integral equation with respect to x using the Leibniz rule, we obtain: f'(x) + f(x) k - ( _e f(x))^2 = 0 f'(x) f(x) k - ( _e f(x))^2 = -1 Integrating both sides with respect to x : f'(x) f(x) k - ( _e f(x))^2 dx = -1 dx Let _e f(x) = u , then f'(x) f(x) dx = du . The integral becomes: du k - u^2 = -x + C ⁻¹ ( u k ) = -x + C ⁻¹ ( _e f(x) k ) = -x + C Using the initial condi

Let k > 0 be a constant. A differentiable function f: [0, 2 ] R ^+ satisfies f(x) + ₀^ x f(t) k - ( _e f(t))^2 dt = e^ k for all x [0, 2 ] . If f ( 3 ) = e^2 , then the value of k is equal to

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