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\(f\left(x\right)=ax^2+bx+c\)
\(f\left(2\right)=4a+2b+c\)
\(f\left(-1\right)=a-b+c\)
\(\Rightarrow f\left(2\right)+f\left(-1\right)=4a+2b+c+a-b+c\)
\(\Leftrightarrow f\left(2\right)+f\left(-1\right)=5a+b+2c=0\)
\(\Rightarrow f\left(2\right)+f\left(-1\right)=0\Leftrightarrow f\left(2\right)=-f\left(-1\right)\)
\(\Leftrightarrow f\left(2\right).f\left(-1\right)=-f\left(-1\right).f\left(-1\right)\le0\)
\(\Rightarrowđpcm\)
\(f\left(-1\right)=a\left(-1\right)^2+b.\left(-1\right)+c\)
\(=a-b+c\)
\(f\left(2\right)=a.2^2+b.2+c\)
\(=4a+2b+c\)
\(\Rightarrow f\left(2\right)-2.f\left(-1\right)=\left(4a+2b+c\right)-2\left(a-b+c\right)\)
\(=2a+4b-c=0\)
\(\Rightarrow f\left(2\right)=2.f\left(-1\right)\)
\(\Rightarrow f\left(2\right)\)và \(2.f\left(-1\right)\)cùng dấu
\(\Rightarrow f\left(2\right)\)và \(f\left(-1\right)\)cùng dấu
\(\Rightarrow f\left(2\right).f\left(-1\right)\ge0\)(đpcm)
Ta có :\(f\left(-1\right)=a.\left(-1\right)^2+b.\left(-1\right)+c=a-b+c\)
\(f\left(2\right)=a.2^2+b.2+c=4a+2b+c\)
\(\implies\) \(f\left(2\right)-2f\left(-1\right)=\left(4a+2b+c\right)-2.\left(a-b+c\right)\)
\(\implies\) \(f\left(2\right)=2.f\left(-1\right)\)
\(\implies\) \(f\left(-1\right).f\left(2\right)=f\left(-1\right).2f\left(-1\right)=f\left(-1\right)^2.2\) \(\geq\) \(0\)
\(\implies\) \(f\left(-1\right).f\left(2\right)\) \(\geq\) \(0\) \(\left(đpcm\right)\)
\(f\left(-1\right)=-a+b-c+d=2\)
\(f\left(0\right)=d=1\)
\(f\left(\frac{1}{2}\right)=\frac{1}{8}a+\frac{1}{4}b+\frac{1}{2}c+d=3\)
\(f\left(1\right)=a+b+c+d=7\)
Suy ra \(\hept{\begin{cases}-a+b-c=1\\\frac{1}{8}a+\frac{1}{4}b+\frac{1}{2}c=2\\a+b+c=6\end{cases}}\Leftrightarrow\hept{\begin{cases}2b=7\\\frac{1}{8}a+\frac{1}{4}b+\frac{1}{2}c=2\\a+b+c=6\end{cases}}\Leftrightarrow\hept{\begin{cases}a=\frac{1}{3}\\b=\frac{7}{2}\\c=\frac{13}{6}\end{cases}}\)
Lời giải:
Ta có:
$f(4)=16a+4b+c$
$f(-2)=4a-2b+c$
Cộng theo vế: $f(4)+f(-2)=20a+2b+2c=2(10a+b+c)=2.0=0$
$\Rightarrow f(-2)=-f(4)$
$\Rightarrow f(4).f(-2)=f(4).-f(4)=-f(4)^2\leq 0$
Ta có đpcm.
Lời giải:
Ta có:
$f(-1)=a-b+c$
$f(2)=4a+2b+c$
Cộng lại ta có: $f(-1)+f(2)=5a+b+2c=0$
$\Rightarrow f(-1)=-f(2)$
$\Rightarrow f(-1)f(2)=-f(2)^2\leq 0$ (đpcm)
\(f\left(-2\right)=a.\left(-2\right)^2+b.\left(-2\right)+c=4a-2b+c\)
\(f\left(3\right)=a.3^2+b.3+c=9a+3b+c\)
\(f\left(-2\right)+f\left(3\right)=13a+b+2c=0\)
\(\Rightarrow f\left(-2\right)=-f\left(3\right)\Rightarrow f\left(-2\right).f\left(3\right)\le0\)
\(f\left(x\right)=ax^2+bx+c\)
\(\Rightarrow f\left(\frac{1}{2}\right)=\frac{1}{4}a+\frac{1}{2}b+c\)
\(\Rightarrow f\left(-2\right)=4a-2b+c\)
\(\Rightarrow f\left(\frac{1}{2}\right)+f\left(-2\right)=\frac{17}{4}a-\frac{3}{2}b+2c\)
\(\Rightarrow4\left[f\left(\frac{1}{2}\right)+f\left(-2\right)\right]=17a-6b+8c=0\)( vì 17a-6b+8c=0)
\(\Rightarrow f\left(\frac{1}{2}\right)+f\left(-2\right)=0\)
\(\Rightarrow f\left(\frac{1}{2}\right)=-f\left(-2\right)\)
\(\Rightarrow f\left(\frac{1}{2}\right).f\left(-2\right)=-\left[f\left(-2\right)\right]^2\le0\left(đpcm\right)\)