Question

The function g(x) = $e ^ { x ^ { 2 } }$ $\frac{\log(\pi + x)}{\log(e + x)}$ ( x ³ 0) is

• A. Increasing on [0, )
• B. Decreasing on [0, )
• C. Increasing on [0, p/e) and decreasing on [p/e, )
• D. Decreasing on [0, p/e) and increasing on [p/e, )

Answer 1

Answer: (B)

Decreasing on [0, )

Answer 2

Since $e^{x^{2}}$increases on [0, ) so it is enough to consider

f(x) = $\frac{\log(\pi + x)}{\log(e + x)}$

f¢(x) =

= $\frac { \log ( e + x ) \times ( e + x ) - ( \pi + x ) \log ( \pi + x ) } { ( \pi + x ) ( e + x ) ( \log ( e + x ) ) ^ { 2 } }$

Since log function is an increasing function and

e < p, log (e + x) < log (p+ x)

Thus (e + x) log (e + x) < (e + x) log (p + x) < (p + x) log (p + x) for all x > 0.

Thus, f¢(x) < 0 for " x > 0 Ž f(x) decreases on (0, )