
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
For a function  $ f(x) \equal{} 6x(1 \minus{} x)$ , suppose that positive constant  $ c$  and a linear function  $ g(x) \equal{} ax \plus{} b\ (a,\ b: \text{constants}\,\ a > 0)$  satisfy the following 3 conditions:  $ c^2\int_0^1 f(x)\ dx \equal{} 1,\ \int_0^1 f(x)\{g(x)\}^2\ dx \equal{} 1,\ \int_0^1 f(x)g(x)\ dx \equal{} 0$ . Answer the following questions.

(1) Find the constants  $ a,\ b,\ c$ .

(2) For natural number  $ n$ , let  $ I_n \equal{} \int_0^1 x^ne^x\ dx$ . Express  $ I_{n \plus{} 1}$  in terms of  $ I_n$ . Then evaluate  $ I_1,\ I_2,\ I_3$ .

(3) Evaluate the definite integrals  $ \int_0^1 e^xf(x)\ dx$  and  $ \int_0^1 e^xf(x)g(x)\ dx$ .

(4) For real numbers  $ s,\ t$ , define  $ J \equal{} \int_0^1 \{e^x \minus{} cs \minus{} tg(x)\}^2\ dx$ . Find the constants  $ A,\ B,\ C,\ D,\ E$  by setting  $ J \equal{} As^2 \plus{} Bst \plus{} Ct^2 \plus{} Ds\plus{}Et \plus{} F$ .
(You don't need to find the constant  $ F$ ).

(5) Find the values of  $ s,\ t$  for which  $ J$  is minimal.

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