Question

at $\alpha = 0 \Rightarrow f(0)$

$${x = 0,x = 1,y^{2} = x }{y = |0 \cdot x - 5| - |1 - 0 \cdot x| + 0 \cdot x^{2} }{y = 4 }$$

$${A_{1} = \int_{0}^{1}\mspace{2mu}(4 - \sqrt{x})dx }{= 4x - \left. \ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right|_{0}^{1} }{= 4 - \frac{2}{3}(1) = \frac{10}{3} }$$at $\alpha = 1 \Rightarrow f(1)$

$x = 0,x = 1,y^{2} = x$,

$y = |x - 5| - |1 - x| + x^{2}$ in $x \in (0,1)$

$${y = 5 - x - (1 - x) + x^{2} }{y = 4 + x^{2} }$$

$${A_{2} = \int_{0}^{1}\mspace{2mu}\left( \left( 4 + x^{2} \right) - (\sqrt{x}) \right)dx }{= 4x + \frac{x^{3}}{3} - \left. \ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right|_{0}^{1} }{= 4 + \frac{1}{3} - \frac{2}{3} = \frac{11}{3} }{|f(0) + f(1)| = \left| A_{1} + A_{2} \right| = \left| \frac{10}{3} + \frac{11}{3} \right| = \left| \frac{21}{3} \right| = 7 }$$option (3)