MHB Minimum Value of F(x): Does It Have a Maximum?

[wonguyen1995]

Find the minimum value of F(x)=\int_{x^2-2x}^{0} 1/(1+t^2)\,d
Does it has a maximum value? why?

 

[chisigma]

Find the minimum value of F(x)=\int_{x^2-2x}^{0} 1/(1+t^2)\,d
Does it has a maximum value? why?

Defining the function as...

$\displaystyle f(x) = \int_{x^{2} - 2 x}^{0} \frac{d t } {1 + t^{2}}\ (1)$

... it is easy to see that is f(x)> 0 for 0 < x < 2 and f(x) < 0 outside this interval. Since the derivative of f(x) outside the interval 0 < x < 2 never vanishes, f(x) don't have minimum. The maximum of the function is the minimum of the function $\displaystyle g(x) = x^{2} - 2\ x$ and it happens for x=1...

Kind regards

$\chi$ $\sigma$