First slide

Monotonicity

Question

Let $f (x) = x \sqrt{4 ax - x^{2}}, (a > 0) .$ Then f(x) is

Moderate

A

increasing in (0, 3a)
B

decreasing in $(5 a, \infty)$
C

increasing in (0,4a)
D

decreasing in (3a, 4a)

Solution

$f (x) = x \sqrt{4 ax - x^{2}} (domain is [0, 4 a]) ∴ f^{'} (x) = \sqrt{4 ax - x^{2}} + \frac{x (4 a - 2 x)}{2 \sqrt{4 ax - x^{2}}} = \frac{2 x (3 a - x)}{\sqrt{4 ax - x^{2}}} Now, if f^{'} (x) > 0, then 2 x (3 a - x) > 0 or 2 x (x - 3 a) < 0 or x \in (0, 3 a) Thus, / (x) increases in (0, 3 a) and decreases in (3 a, 4 a) .$

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