Cho \(f\left( x \right) = \dfrac{1}{{{x^2} - 4x + 5}} - \dfrac{{{x^2}}}{4} + x\). Gọi \(M = \mathop {Max}\limits_{x \in \left[ {0;3} \right]} f\left( x \right);\) \(m = \mathop {Min}\limits_{x \in \left[ {0;3} \right]} f\left( x \right).\) Khi đó\(M-m\) bằng:

Ta có :

\[\begin{array}{l}f\left( x \right) = \dfrac{1}{{{x^2} - 4x + 5}} - \dfrac{{{x^2}}}{4} + x\\f\left( x \right) = \dfrac{1}{{{x^2} - 4x + 5}} - \dfrac{{{x^2} - 4x}}{4}\end{array}\]

Đặt \(t = {x^2} - 4x + 5\) với \(x \in \left[ {0;3} \right]\) ta có \(t' = 2x - 4 = 0 \Leftrightarrow x = 2 \in \left[ {0;3} \right]\).

Ta có : \(t\left( 0 \right) = 5;\,\,t\left( 2 \right) = 1,\,\,t\left( 3 \right) = 2\).

 \( \Rightarrow \) Với \(x \in \left[ {0;3} \right]\) thì \(t \in \left[ {1;5} \right]\), khi đó hàm số trở thành \(f\left( t \right) = \dfrac{1}{t} - \dfrac{{t - 5}}{4}\) với \(t \in \left[ {1;5} \right]\).

Ta có \(f'\left( t \right) =  - \dfrac{1}{{{t^2}}} - \dfrac{1}{4} < 0\,\,\forall t \in \left[ {1;5} \right]\).

\( \Rightarrow \) Hàm số \(y = f\left( t \right)\) nghịch biến trên \(\left[ {1;5} \right]\) \( \Rightarrow \left\{ \begin{array}{l}\mathop {\max }\limits_{\left[ {0;3} \right]} f\left( x \right) = \mathop {\max }\limits_{\left[ {1;5} \right]} f\left( t \right) = f\left( 1 \right) = 2 = M\\\mathop {\min }\limits_{\left[ {0;3} \right]} f\left( x \right) = \mathop {\min }\limits_{\left[ {1;5} \right]} f\left( t \right) = f\left( 5 \right) = \dfrac{1}{5} = m\end{array} \right.\)

Vậy \(M - m = 2 - \dfrac{1}{5} = \dfrac{9}{5}\).