Let f(x) = ( k e^2 x )^ x^2 for x > 0 , where k is a positive constant. If the maximum value of f(x) is e^ 8 e^3 , then the value of k is

Step-by-step solution

Let y = f(x) = ( k e^2 x )^ x^2 . Taking the natural logarithm on both sides, we get: y = x^2 ( k + 2 - x) Differentiating with respect to x : 1 y dy dx = 2x ( k + 2 - x) + x^2 (- 1 x ) dy dx = y [ 2x k + 4x - 2x x - x ] dy dx = y x (2 k + 3 - 2 x) For a local maximum or minimum, dy dx = 0 . Since y > 0 and x > 0 , we have: 2 k + 3 - 2 x = 0 x = k + 3 2 x = k e^ 3/2 To find the maximum value, substitute x = k e^ 3/2 into y : y_ = (k e^ 3/2 )^2 ( k + 2 - ( k + 3 2 ) ) y_ = k^2 e^3 ( 1 2 ) = 1 2 k^2 e^3 Given that th