
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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)$ .

Present the answer in LaTex format: \boxed{Your answer}
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