f(x)= 2x²-7x+1

\(=2\left(x^2-\frac{7x}{2}+\frac{1}{2}\right)\)

\(=2\left(x^2-\frac{7x}{2}+\frac{49}{16}\right)-\frac{41}{8}\)

\(=2\left(x-\frac{7}{4}\right)^2-\frac{41}{8}\ge0-\frac{41}{8}=-\frac{41}{8}\)

Dấu = khi \(2\left(x-\frac{7}{4}\right)^2=0\Leftrightarrow x-\frac{7}{4}\Leftrightarrow x=\frac{7}{4}\)

Vậy...

1,\(f\left(x\right)=3x^2-2x-7\)

\(=3\left(x^2-\dfrac{2}{3}x+\dfrac{1}{9}\right)-\dfrac{22}{3}\)

\(=2\left(x-\dfrac{1}{3}\right)^2-\dfrac{22}{3}\ge-\dfrac{22}{3}\forall x\)

Vậy GTNN của biểu thức là \(-\dfrac{22}{3}\) khi \(x-\dfrac{1}{3}=0\Rightarrow x=\dfrac{1}{3}\)

\(b,f\left(x\right)=5x^2+7x=5\left(x^2+\dfrac{7}{5}x+\dfrac{49}{100}\right)-\dfrac{49}{20}\)\(=5\left(x+\dfrac{7}{10}\right)^2-\dfrac{49}{20}\ge-\dfrac{49}{20}\forall x\)

Vậy Giá trị nhỏ nhất của biểu thức là \(-\dfrac{49}{20}\) khi \(x+\dfrac{7}{10}=0\Rightarrow x=-\dfrac{7}{10}\)

\(c,f\left(x\right)=-5x^2+9x-2=-5\left(x^2-\dfrac{9}{5}x+\dfrac{81}{100}\right)+\dfrac{41}{20}\)\(=-5\left(x-\dfrac{9}{10}\right)^2+\dfrac{41}{20}\le\dfrac{41}{20}\forall x\)

Vậy GTLN của biểu thức là \(\dfrac{41}{20}\) khi \(x-\dfrac{9}{10}=0\Rightarrow x=\dfrac{9}{10}\)

\(d,f\left(x\right)=-7x^2+3x=-7\left(x^2-\dfrac{3}{7}x+\dfrac{9}{196}\right)+\dfrac{9}{28}\)\(=-7\left(x-\dfrac{3}{14}\right)^2+\dfrac{9}{28}\le\dfrac{9}{28}\forall x\)

Vậy GTLN của biểu thức là \(\dfrac{9}{28}\) khi \(x-\dfrac{3}{14}=0\Rightarrow x=\dfrac{3}{14}\)

1/ \(f\left(x\right)=3x^2-2x-7\)

\(=3\left(x^2-\dfrac{2}{3}x-7\right)\)

\(=3\left(x^2-\dfrac{2}{3}+\dfrac{1}{9}-\dfrac{64}{9}\right)\)

\(=3\left(x-\dfrac{1}{3}\right)^2-\dfrac{64}{3}\)

Ta có: \(3\left(x-\dfrac{1}{3}\right)^2\ge0\forall x\Rightarrow3\left(x-\dfrac{1}{3}\right)^2-\dfrac{64}{3}\ge-\dfrac{64}{3}\forall x\)

Dấu "=" xảy ra khi \(x-\dfrac{1}{3}=0\) hay \(x=\dfrac{1}{3}\)

Vậy MIN_f(x) = \(-\dfrac{64}{3}\) khi x = \(\dfrac{1}{3}\).

2/ \(f\left(x\right)=5x^2+7x\)

\(=5\left(x^2+\dfrac{7}{5}x\right)=5\left(x^2+\dfrac{7}{5}x+\dfrac{49}{100}-\dfrac{49}{100}\right)\)

\(=5\left(x+\dfrac{7}{10}\right)^2-\dfrac{49}{20}\)

Ta có: \(5\left(x+\dfrac{7}{10}\right)^2\ge0\forall x\Rightarrow5\left(x+\dfrac{7}{10}\right)^2-\dfrac{49}{20}\ge-\dfrac{49}{20}\forall x\)

Dấu "=" xảy ra khi \(x+\dfrac{7}{10}=0\) hay \(x=-\dfrac{7}{10}\)

Vậy MIN_f(x) = \(-\dfrac{49}{20}\) khi x = \(-\dfrac{7}{10}\).

1/ \(f\left(x\right)=-5x^2+9x-2\)

\(=-5\left(x^2-\dfrac{9}{5}x+\dfrac{2}{5}\right)\)

\(=-5\left(x^2-\dfrac{9}{5}x+\dfrac{81}{100}-\dfrac{41}{100}\right)\)

\(=-5\left(x-\dfrac{9}{10}\right)^2+\dfrac{41}{20}\)

Ta có: \(-5\left(x-\dfrac{9}{10}\right)^2\le0\forall x\Rightarrow-5\left(x-\dfrac{9}{10}\right)^2+\dfrac{41}{20}\le\dfrac{41}{20}\forall x\)

Dấu "=" xảy ra khi \(x-\dfrac{9}{10}=0\) hay \(x=\dfrac{9}{10}\)

Vậy MAX_f(x) = \(\dfrac{41}{20}\) khi x = \(\dfrac{9}{10}\)

2/ \(f\left(x\right)=-7x^2+3x=-7\left(x^2-\dfrac{3}{7}x+\dfrac{9}{196}\right)+\dfrac{9}{28}\)

\(=-7\left(x-\dfrac{3}{14}\right)^2+\dfrac{9}{28}\)

Ta có: \(-7\left(x-\dfrac{3}{14}\right)^2\le0\forall x\Rightarrow-7\left(x-\dfrac{3}{14}\right)^2+\dfrac{9}{28}\le\dfrac{9}{28}\forall x\)

Dấu "=" xảy ra khi \(x-\dfrac{3}{14}=0\) hay x = \(\dfrac{3}{14}\)

Vậy MAX_f(x) = \(\dfrac{9}{28}\) khi x = \(\dfrac{3}{14}\).

1: \(\dfrac{f\left(x\right)}{x-3}=\dfrac{2x^2-6x+\left(a+6\right)x-3a-18+3a+19}{x-3}\)

=2x^2+(a+6)+3a+19/x-3

Để f(x)/x-3 dư 4 thì 3a+19=4

=>3a=-15

=>a=-5

2: \(\dfrac{f\left(x\right)}{x-5}=\dfrac{3x^2-15x+\left(a+15\right)x-5a-75+5a+102}{x-5}\)

\(=3x+a+15+\dfrac{5a+102}{x-5}\)

Để dư là 27 thì 5a+102=27

=>5a=-75

=>a=-15

\(f\left(x\right)=\left|x-2015\right|+\left|x+2016\right|\)

a) Ta có: \(\left|x\right|=\orbr{\begin{cases}x=\frac{1}{2}\\x=-\frac{1}{2}\end{cases}}\)

+) Với \(x=\frac{1}{2}\): 

\(f\left(\frac{1}{2}\right)=\left|\frac{1}{2}-2015\right|+\left|\frac{1}{2}+2016\right|=2\)

+) Với \(x=-\frac{1}{2}\)

\(f\left(-\frac{1}{2}\right)=\left|-\frac{1}{2}-2015\right|+\left|-\frac{1}{2}+2016\right|=0\)

c) Áp dụng BĐT |x| + |y| \(\ge\)|x + y|, ta được:

\(f\left(x\right)=\left|x-2015\right|+\left|x+2016\right|=\left|2015-x\right|+\left|x+2016\right|\)

\(\ge\left|\left(2015-x\right)+\left(x+2016\right)\right|=\left|4031\right|=4031\)

(Dấu "="\(\Leftrightarrow\left(2015-x\right)\left(x+2016\right)\ge0\)

TH1: \(\hept{\begin{cases}2015-x\ge0\\x+2016\ge0\end{cases}}\Leftrightarrow-2016\le x\le2015\)

TH2: \(\hept{\begin{cases}2015-x\le0\\x+2016\le0\end{cases}}\Leftrightarrow\hept{\begin{cases}x\ge2015\\x\le-2016\end{cases}}\left(L\right)\))

Vậy \(f\left(x\right)_{min}=4031\Leftrightarrow-2016\le x\le2015\)