Consider the following function $f(x)$. $\displaystyle f(x) = \int_0^1 \frac{|t-x|}{1+t^2}\,dt \quad (0 \leq x \leq 1)$
(1) Find all real numbers $\alpha$ satisfying $0 < \alpha < \dfrac{\pi}{4}$ such that $f'(\tan\alpha) = 0$.
(2) For the value of $\alpha$ found in (1), find the value of $\tan\alpha$.
(3) Find the maximum value and the minimum value of $f(x)$ on the interval $0 \leq x \leq 1$. If necessary, you may use the fact that $0.69 < \log 2 < 0.7$. %% Page 3
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Consider the following function $f(x)$. \quad $\displaystyle f(x) = \int_0^1 \frac{|t-x|}{1+t^2}\,dt \quad (0 \leq x \leq 1)$
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\noindent(1) \quad Find all real numbers $\alpha$ satisfying $0 < \alpha < \dfrac{\pi}{4}$ such that $f'(\tan\alpha) = 0$.
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\noindent(2) \quad For the value of $\alpha$ found in (1), find the value of $\tan\alpha$.
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\noindent(3) \quad Find the maximum value and the minimum value of $f(x)$ on the interval $0 \leq x \leq 1$. If necessary, you may use the fact that $0.69 < \log 2 < 0.7$.
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