MathDB

Let f(x) \equal{} e^{(p \plus{} 1)x} \minus{} e^x for real number

p > 0

. Answer the following questions. (1) Find the value of x \equal{} s_p for which

f(x)

is minimal and draw the graph of y \equal{} f(x). (2) Let g(t) \equal{} \int_t^{t \plus{} 1} f(x)e^{t \minus{} x}\ dx. Find the value of t \equal{} t_p for which

g(t)

is minimal. (3) Use the fact 1 \plus{} \frac {p}{2}\leq \frac {e^p \minus{} 1}{p}\leq 1 \plus{} \frac {p}{2} \plus{} p^2\ (0 < p\leq 1) to find the limit \lim_{p\rightarrow \plus{}0} (t_p \minus{} s_p).

Problem Statement