Tìm GTNN của :
F = ( x2 + 5x + 5 )[( x + 2 )( x + 3 ) + 1]
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4 tháng 10 2021
a) \(4x^2+12x+1=\left(4x^2+12x+9\right)-8=\left(2x+3\right)^2-8\ge-8\)
\(ĐTXR\Leftrightarrow x=-\dfrac{3}{2}\)
b) \(4x^2-3x+10=\left(4x^2-3x+\dfrac{9}{16}\right)+\dfrac{151}{16}=\left(2x-\dfrac{3}{4}\right)^2+\dfrac{151}{16}\ge\dfrac{151}{16}\)
\(ĐTXR\Leftrightarrow x=\dfrac{3}{8}\)
c) \(2x^2+5x+10=\left(2x^2+5x+\dfrac{25}{8}\right)+\dfrac{55}{8}=\left(\sqrt{2}x+\dfrac{5\sqrt{2}}{4}\right)^2+\dfrac{55}{8}\ge\dfrac{55}{8}\)
\(ĐTXR\Leftrightarrow x=-\dfrac{5}{4}\)
d) \(x-x^2+2=-\left(x^2-x+\dfrac{1}{4}\right)+\dfrac{9}{4}=-\left(x-\dfrac{1}{2}\right)^2+\dfrac{9}{4}\le\dfrac{9}{4}\)
\(ĐTXR\Leftrightarrow x=\dfrac{1}{2}\)
e) \(2x-2x^2=-2\left(x^2-x+\dfrac{1}{4}\right)+\dfrac{1}{2}=-2\left(x-\dfrac{1}{2}\right)^2+\dfrac{1}{2}\le\dfrac{1}{2}\)
\(ĐTXR\Leftrightarrow x=\dfrac{1}{2}\)
f) \(4x^2+2y^2+4xy+4y+5=\left(4x^2+4xy+y^2\right)+\left(y^2+4y+4\right)+1=\left(2x+y\right)^2+\left(y+2\right)^2+1\ge1\)
\(ĐTXR\Leftrightarrow\) \(\left\{{}\begin{matrix}x=1\\y=-2\end{matrix}\right.\)
4 tháng 10 2021
a: Ta có: \(4x^2+12x+1\)
\(=4x^2+12x+9-8\)
\(=\left(2x+3\right)^2-8\ge-8\forall x\)
Dấu '=' xảy ra khi \(x=-\dfrac{3}{2}\)
b: Ta có: \(4x^2-3x+10\)
\(=4\left(x^2-\dfrac{3}{4}x+\dfrac{5}{2}\right)\)
\(=4\left(x^2-2\cdot x\cdot\dfrac{3}{8}+\dfrac{9}{64}+\dfrac{151}{64}\right)\)
\(=4\left(x-\dfrac{3}{8}\right)^2+\dfrac{151}{16}\ge\dfrac{151}{16}\forall x\)
Dấu '=' xảy ra khi \(x=\dfrac{3}{8}\)
c: Ta có: \(2x^2+5x+10\)
\(=2\left(x^2+\dfrac{5}{2}x+5\right)\)
\(=2\left(x^2+2\cdot x\cdot\dfrac{5}{4}+\dfrac{25}{16}+\dfrac{55}{16}\right)\)
\(=2\left(x+\dfrac{5}{4}\right)^2+\dfrac{55}{8}\ge\dfrac{55}{8}\forall x\)
Dấu '=' xảy ra khi \(x=-\dfrac{5}{4}\)
4 tháng 1 2022
Bài 2:
a: \(\Leftrightarrow2x^2-10x-3x-2x^2=26\)
=>-13x=26
hay x=-2
b: \(\Leftrightarrow\left(x-1\right)\left(5x-1\right)=0\)
hay \(x\in\left\{1;\dfrac{1}{5}\right\}\)
c: \(\Leftrightarrow\left(x+5\right)\left(2-x\right)=0\)
hay \(x\in\left\{-5;2\right\}\)
28 tháng 7 2023
1:
a: =x^2-7x+49/4-5/4
=(x-7/2)^2-5/4>=-5/4
Dấu = xảy ra khi x=7/2
b: =x^2+x+1/4-13/4
=(x+1/2)^2-13/4>=-13/4
Dấu = xảy ra khi x=-1/2
e: =x^2-x+1/4+3/4=(x-1/2)^2+3/4>=3/4
Dấu = xảy ra khi x=1/2
f: x^2-4x+7
=x^2-4x+4+3
=(x-2)^2+3>=3
Dấu = xảy ra khi x=2
2:
a: A=2x^2+4x+9
=2x^2+4x+2+7
=2(x^2+2x+1)+7
=2(x+1)^2+7>=7
Dấu = xảy ra khi x=-1
b: x^2+2x+4
=x^2+2x+1+3
=(x+1)^2+3>=3
Dấu = xảy ra khi x=-1
VT
2
5 tháng 4 2018
a)
\(f\left(x\right)=3x^2-5x+1\)
\(3f\left(x\right)=9x^2-15x+3\)
\(3f\left(x\right)=\left(9x^2-15x+\frac{25}{4}\right)-\frac{13}{4}\)
\(3f\left(x\right)=\left(3x-\frac{5}{2}\right)^2-\frac{13}{4}\)
Mà \(\left(3x-\frac{5}{2}\right)^2\ge0\forall x\)
\(\Rightarrow3f\left(x\right)\ge\frac{-13}{4}\)
\(\Leftrightarrow f\left(x\right)\ge-\frac{13}{12}\)
Dấu '=' xảy ra khi :
\(3x-\frac{5}{2}=0\Leftrightarrow3x=\frac{5}{2}\Leftrightarrow x=\frac{5}{6}\)
5 tháng 4 2018
\(f\left(x\right)=2x^2-9x-3\)
\(2f\left(x\right)=4x^2-18x-6\)
\(2f\left(x\right)=\left(4x^2-18x+\frac{81}{4}\right)-\frac{105}{4}\)
\(2f\left(x\right)=\left(2x-\frac{9}{2}\right)^2-\frac{105}{4}\)
Mà \(\left(2x-\frac{9}{2}\right)^2\ge0\forall x\)
\(\Rightarrow2f\left(x\right)\ge-\frac{105}{4}\)
\(\Leftrightarrow f\left(x\right)\ge-\frac{105}{8}\)
Dấu "=" xảy ra khi :
\(2x-\frac{9}{2}=0\Leftrightarrow2x=\frac{9}{2}\Leftrightarrow x=\frac{9}{4}\)
26 tháng 7 2021
1) \(\left(\dfrac{1}{2}x+3\right)\left(x^2-4x-6\right)\)
\(=\dfrac{1}{2}x^3-2x^2-3x+3x^2-12x-18\)
\(=\dfrac{1}{2}x^3+x^2-15x-18\)
2) \(\left(6x^2-9x+15\right)\left(\dfrac{2}{3}x+1\right)\)
\(=4x^3+6x^2-6x^2-9x+10x+15\)
\(=4x^3+x+15\)
3) Ta có: \(\left(3x^2-x+5\right)\left(x^3+5x-1\right)\)
\(=3x^5+15x^2-3x^2-x^4-5x^2+x+5x^3+25x-5\)
\(=3x^5-x^4+5x^3+10x^2+26x-5\)
4) Ta có: \(\left(x-1\right)\left(x+1\right)\left(x-2\right)\)
\(=\left(x^2-1\right)\left(x-2\right)\)
\(=x^3-2x^2-x+2\)
F = \(\left[\left(x+2\right)\left(x+3\right)-1\right]\) . \(\left[\left(x+2\right)\left(x+3\right)+1\right]\)
= \(\left(x+2\right)^2\left(x+3\right)^2-1\) \(\ge-1\)
Dấu "=" xảy ra <=> \(\left(x+2\right)^2\left(x+3\right)^2\) =0
<=> (x+2)(x+3) = 0
<=> \(\left[{}\begin{matrix}x=-2\\x=-3\end{matrix}\right.\)
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