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để f(x)>0 với mọi x thì:
\(\left\{{}\begin{matrix}\Delta'< 0\\a>0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left(m-2\right)^2-2\left(m^2+2\right)< 0\\m^2+2>0\left(lđ\right)\end{matrix}\right.\)\(\Leftrightarrow m^2+4m>0\Leftrightarrow\left[{}\begin{matrix}m>0\\m< -4\end{matrix}\right.\)
\(f\left(x\right)>0\forall x\in R\)
\(\Rightarrow\left\{{}\begin{matrix}m^2+2>0\left(LĐ\right)\\\Delta'=\left[-\left(m-1\right)\right]^2-\left(m^2+2\right)\cdot1< 0\end{matrix}\right.\)
\(\Rightarrow m^2-2m+1-m^2-2< 0\)
\(\Leftrightarrow m>-\dfrac{1}{2}\)
Vậy: \(m>-\dfrac{1}{2}\).
\(f\left(x\right)=\left(m-4\right)x^2+\left(m+1\right)x+2m-1\)
\(f\left(x\right)< 0,\forall x\in R\Leftrightarrow\left\{{}\begin{matrix}a< 0\\\Delta< 0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m-4< 0\\\left(m+1\right)^2-4\left(m-4\right)\left(2m-1\right)< 0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m< 4\\m^2+2m+1-4\left(2m^2-m-8m+4\right)< 0\end{matrix}\right.\)
\(\Leftrightarrow m^2+2m+1-8m^2+36m-16< 0\)
\(\Leftrightarrow-7m^2+38m-15< 0\)
\(\Leftrightarrow\left\{{}\begin{matrix}m< 4\\\left[{}\begin{matrix}m< \dfrac{3}{7}\\m>5\end{matrix}\right.\end{matrix}\right.\)
\(KL:m\in\left(5;+\infty\right)\)
Do \(a=1>0\) nên \(f\left(x\right)\ge0;\forall x\) khi:
\(\Delta'=\left(m-3\right)^2-4m\le0\)
\(\Leftrightarrow m^2-10m+9\le0\)
\(\Rightarrow1\le m\le9\)
Đáp án: D
A: \(f(x)=x^2+2x-x^2=2x\) → Bậc 1.
B: \(f(x)=x+3\) → Bậc 1.
C: Bậc 4.
\(f\left(x\right)>0,\forall x\in R\Leftrightarrow\left\{{}\begin{matrix}a>0\\\Delta< 0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m+1>0\\\left[-2\left(m-1\right)\right]^2-4\left(m+1\right)\left(-m+4\right)< 0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m>-1\\4\left(m^2-2m+1\right)-4\left(-m^2+4m-m+4\right)< 0\end{matrix}\right.\)
\(\Leftrightarrow4m^2-8m+4+4m^2-12m-16< 0\)
\(\Leftrightarrow8m^2-20m-12< 0\)
\(KL:m\in\left(-1;3\right)\)
Để \(f\left(x\right)>0\Rightarrow\Delta'>0\Rightarrow\left(m-2\right)^2-2\left(m^2+2\right)>0\Leftrightarrow m^2-4m+4-2m^2-4>\Leftrightarrow-m^2-4m>0\Leftrightarrow m^2+4m< 0\Leftrightarrow m\left(m+4\right)< 0\Leftrightarrow-4< m< 0\)