
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
Consider a function  $ f(x)\equal{}xe^{\minus{}x^3}$  defined on any real numbers.

(1) Examine the variation and convexity of  $ f(x)$  to draw the garph of  $ f(x)$ .

(2) For a positive number  $ C$ , let  $ D_1$  be the region bounded by  $ y\equal{}f(x)$ , the  $ x$ -axis and  $ x\equal{}C$ . Denote  $ V_1(C)$  the volume obtained by rotation of  $ D_1$  about the  $ x$ -axis. Find  $ \lim_{C\rightarrow \infty} V_1(C)$ .

(3) Let  $ M$  be the maximum value of  $ y\equal{}f(x)$  for  $ x\geq 0$ . Denote  $ D_2$  the region bounded by  $ y\equal{}f(x)$ , the  $ y$ -axis and  $ y\equal{}M$ .
Find the volume  $ V_2$  obtained by rotation of  $ D_2$  about the  $ y$ -axis.

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