gaokao 2019 Q19

gaokao · China · beijing-science 14 marks Stationary points and optimisation Find absolute extrema on a closed interval or domain

Given the function $f ( x ) = \frac { 1 } { 4 } x ^ { 3 } - x ^ { 2 } + x$. (I) Find the equation of the tangent line to the curve $y = f ( x )$ with slope 1; (II) When $x \in [ - 2,4 ]$, prove that: $x - 6 \leqslant f ( x ) \leqslant x$; (III) Let $F ( x ) = | f ( x ) - ( x + a ) | ( a \in \mathbb { R } )$. Let $M ( a )$ denote the maximum value of $F ( x )$ on the interval $[ - 2,4 ]$. When $M ( a )$ is minimized, find the value of $a$.

Solution:

Given the function $f ( x ) = \frac { 1 } { 4 } x ^ { 3 } - x ^ { 2 } + x$.
(I) Find the equation of the tangent line to the curve $y = f ( x )$ with slope 1;
(II) When $x \in [ - 2,4 ]$, prove that: $x - 6 \leqslant f ( x ) \leqslant x$;
(III) Let $F ( x ) = | f ( x ) - ( x + a ) | ( a \in \mathbb { R } )$. Let $M ( a )$ denote the maximum value of $F ( x )$ on the interval $[ - 2,4 ]$. When $M ( a )$ is minimized, find the value of $a$.

Paper Questions

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