A real valued function f(x) = |x^2 - 3x + 2| | vedclass

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(D) Given $f(x) = |x^2 - 3x + 2| + 2x - 3$ on $[-2, 1]$.Since $x^2 - 3x + 2 = (x - 1)(x - 2)$,for $x \in [-2, 1]$,$(x - 1) \le 0$ and $(x - 2) < 0$,so

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(D) Given $f(x) = |x^2 - 3x + 2| + 2x - 3$ on $[-2, 1]$.Since $x^2 - 3x + 2 = (x - 1)(x - 2)$,for $x \in [-2, 1]$,$(x - 1) \le 0$ and $(x - 2) Thus,$|x^2 - 3x + 2| = x^2 - 3x + 2$.Substituting this into $f(x)$,we get $f(x) = x^2 - 3x + 2 + 2x - 3 = x^2 - x - 1$.To find the extrema on $[-2, 1]$,we check the critical points and endpoints.$f'(x) = 2x - 1$. Setting $f'(x) = 0$ gives $x = 1/2$,which is in $[-2, 1]$.Calculate values at critical point and endpoints:$f(-2) = (-2)^2 - (-2) - 1 = 4 + 2 - 1 = 5$.$f(1) = (1)^2 - (1) - 1 = -1$.$f(1/2) = (1/2)^2 - (1/2) - 1 = 1/4 - 1/2 - 1 = -5/4$.Comparing these,the absolute maximum $M = 5$ and the absolute minimum $m = -5/4$.Then $M - 4m = 5 - 4(-5/4) = 5 + 5 = 10$.

$A$ real valued function $f(x) = |x^2 - 3x + 2| + 2x - 3$ is defined on $[-2, 1]$. If $m$ and $M$ are absolute minimum and absolute maximum values of $f$ respectively,then $M - 4m =$

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