Problem

Problem

Let

f(x) = e^{(p+1)x} - e^x

for real number

p > 0

, then a. The value of

x = s_p

for which

f(x)

is minimum, is b. Let

g(t) = \int_t^{t+1} f(x)e^{t-x} dx

. The value of

t = t_p

, for which

g(t)

is minimum, is c. Use the fact that

1 + \frac{p}{2} \le \frac{e^p - 1}{p} \le 1 + \frac{p}{2} + p^2

(

0 < p \le 1

), the value of

\lim_{p \to 0^+} (s_p - t_p)

is

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