Let =\boldsymbol{x} \sqrt{\mathbf{4} \boldsymbol{a} \boldsymbol{x}-\boldsymbol{x}^{2}},(\boldsymbol{a}>\mathbf{0}) . . \) Then \) is decreasing in: This question has multiple correct options

Maths · Applications of derivatives: Rate of change of quantities, monotonic-Increasing and decreasing functions, Maxima and minima of functions of one variable

Let =\boldsymbol{x} \sqrt{\mathbf{4} \boldsymbol{a} \boldsymbol{x}-\boldsymbol{x

Let \( \boldsymbol{f}(\boldsymbol{x})=\boldsymbol{x} \sqrt{\mathbf{4} \boldsymbol{a} \boldsymbol{x}-\boldsymbol{x}^{2}},(\boldsymbol{a}>\mathbf{0}) . . \) Then \( f(x) \) is decreasing in: This question has multiple correct options

A. \( (5 a, \infty) \) )
B. \( (-\infty, 0) U(4 a, \infty) \)
C. Always increasing
D. None of the above

Step-by-step solution

The function f(x) = x sqrt(4ax - x^2) has domain [0,4a]. Its derivative f'(x) = x(6a - 2x)/sqrt(4ax - x^2). f'(x) > 0 on (0,3a) and f'(x) < 0 on (3a,4a). Thus f is decreasing on (3a,4a). None of the given intervals match this interval; options A and B include points outside the domain, and option C is false. Hence the correct choice is D.

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Step-by-step solution

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