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Recently, I have done some exercise about single variable calculus. Most of them are solved by theorems like Lagrange/Cauchy mean value theorem, Rolle theorem, and Taylor series. And most of the time I can't solve the question if I haven't seen similar ones before. For example, $f(x)$ has second-order derivative in $[a,b]$, show that $\exists$ $\eta\in(a,b)$ satisfy $$\int_a^bf(t)dt=f\left(\frac{a+b}{2}\right)(b-a)+\frac{1}{24}f''(\eta)(b-a)^3$$ I knew this question can be solved by using Taylor's theorem. But when I actually began to do it I lost my way. After I read the answer, the key idea is introduce a function $$g(x)=\int_{x_0}^{x}f(t)dt$$ Use Taylor's theorem $$g(x)=g(x_0)+g'(x_0)(x-x_0)+\frac{1}{2}g''(x_0)(x-x_0)^2+\frac{1}{3!}g'''(\theta)(x-x_0)^3$$ Let $x_0=\frac{a+b}{2}$ then $g(a)-g(b)$ we have$$\int_a^bf(t)dt=f\left(\frac{a+b}{2}\right)(b-a)+\frac{1}{48}[f''(\eta_1)+f''(\eta_2)](b-a)^3$$ Take $f''(\eta)=\frac{f''(\eta_1)+f''(\eta_2)}{2}$ will get what we need.

My question is how one get his/her ideas when meeting with questions like this one. Of course, practice is important, so does observation and patience. Each time I stuck in a question, I read the answer or ask for help and try to understand the key point of the question. I am afraid of this kind of questions, because most of the time I can't solve it and I become a little frustrated. Actually, I read this question myself and I even don't know what kind of advice am I asking for. So, any advice-you think is helpful-are welcome.

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    $\begingroup$ So you are asking "How am I supposed to have ideas that the author seems to draw out of nowhere?"? Mini-answer: Sometimes it really just is like that, the idea popped into the authors mind using his 'mathematical creativity' which usually is - as you beautifully said - mainly based on practice, observation and patience. But I don't think this should depress you since those are qualities which are acquired over time. In contrast to creativity, there often is a concrete motivation behind a theorem which yields useful ideas. But I myself am interested in how others will answer this. $\endgroup$ – Piwi Oct 11 '15 at 17:52

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