First slide

Monotonicity

Question

$Let f (x) = \{\begin{array}{rr} x e^{a x}; x \leq 0 \\ x + a x^{2} - x^{3}; x > 0 \end{array} where a' is a positive constant. Find$ $the interval in which f^{'} (x) is increasing.$

Moderate

A

$[- \frac{2}{a}, \frac{a}{3}]$
B

$(- \frac{2}{a}, \frac{a}{3})$
C

$(\frac{2}{a}, - \frac{- a}{3})$
D

$[\frac{2}{a}, \frac{- a}{3}]$

Solution

$f^{'} (x) = \{\begin{cases} e^{a x} + a x e^{a x}; x \leq 0 \\ 1 + 2 a x - 3 x^{2}; x > 0 \end{cases}$
$\begin{array}{l} f^{''} (x) = \{\begin{cases} 2 a e^{a x} + a^{2} x e^{a x} & ; x \leq 0 \\ 2 a - 6 x & ; x > 0 \end{cases} \\ For f^{'} (x) to be an increasing function f^{''} (x) \geq 0 \\ i.e., a e^{a x} (2 + a x) \geq 0 \Rightarrow x \geq - \frac{2}{a} \end{array}$
$2 a - 6 x \geq 0 \Rightarrow x \leq \frac{a}{3}$
$∴ The int erval inwhich f^{'} (x) is increasing is [- \frac{2}{a}, \frac{a}{3}]$

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