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$The function g (x) = e^{x^{2}} \frac{\log (π + x)}{\log (e + x)} (x \geq 0) is$

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a

increasing on [0,∞)

b

decreasing on [0,∞)

c

increasing on [0,π/e) and decreasing on [π/e,∞)

d

decreasing on [0,π/e,) and increasing on [π/e,∞)

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detailed solution

Correct option is B

Since ex2 increases on [0,∞) so it is enough to consider f(x)=log⁡(π+x)log⁡(e+x)f′(x)=log⁡(e+x)×1π+x−log⁡(π+x)1e+x(log⁡(e+x))2=f′(x)=log⁡(e+x)×(e+x)−(π+x)log⁡(π+x)(π+x)(e+x)(log⁡(e+x))2Since log function is an increasing function and e<π,log⁡(e+x)0 Thus f′(x)<0 for ∀x>0⇒f(x) decreases on (0,∞)

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