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

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−2a , a3

2a , −−a3

2a , −a3

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Correct option is A

f′(x)=eax+axeax ;x≤01+2ax−3x2;x>0f′′(x)=2aeax+a2xeax;x≤02a−6x;x>0 For f′(x) to be an increasing function f′′(x)≥0 i.e., aeax(2+ax)≥0⇒x≥−2a2a−6x≥0⇒x≤a3∴ The int erval inwhich f′(x) is increasing is −2a,a3

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Let f(x)=xeax; x≤0x+ax2−x3; x>0 where a ' is a positive constant. Find the interval in which f′(x) is increasing.

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