Google
Câu hỏi: Cho hàm số $f\left( x \right)={{x}^{3}}-3x+m+2$. Có bao nhiêu số nguyên dương $m\le 50$ sao cho với mọi bộ ba số thực $a,b,c\in \left[ -1;3 \right]$ thì $f\left( a \right),f\left( b \right),f\left( c \right)$ là độ dài ba cạnh một tam giác nhọn?
A. 0.
B. 5.
C. 2.
D. 1.
A. 0.
B. 5.
C. 2.
D. 1.
Đặt $g\left( x \right)={{x}^{3}}-3x+2\Rightarrow \underset{\left[ -1;3 \right]}{\mathop{\max }} g\left( x \right)=20;\underset{\left[ -1;3 \right]}{\mathop{\min }} g\left( x \right)=0$. Khi đó $f\left( x \right)=m+g\left( x \right)$.
Ta có: $\begin{aligned}
& f\left( a \right)+f\left( b \right)>f\left( c \right),\forall a,b,c\in \left[ -1;3 \right]\Rightarrow m>g\left( c \right)-\left( g\left( a \right)+g\left( b \right) \right),\forall a,b,c\in \left[ -1;3 \right] \\
& \Rightarrow m>\underset{\left[ -1;3 \right]}{\mathop{\max }} g\left( x \right)-2\underset{\left[ -1;3 \right]}{\mathop{\min }} g\left( x \right)\Rightarrow m>20. \\
\end{aligned}$
$\begin{aligned}
& {{f}^{2}}\left( a \right)+{{f}^{2}}\left( b \right)>{{f}^{2}}\left( c \right),\forall a,b,c\in \left[ -1;3 \right]\Rightarrow {{\left( m+g\left( a \right) \right)}^{2}}+{{\left( m+g\left( b \right) \right)}^{2}}>{{\left( m+g\left( c \right) \right)}^{2}},\forall a,b,c\in \left[ -1;3 \right] \\
& \Rightarrow {{m}^{2}}+2\left( g\left( a \right)+g\left( b \right)-g\left( c \right) \right)m+{{g}^{2}}\left( a \right)+{{g}^{2}}\left( b \right)-{{g}^{2}}\left( c \right)>0,\forall a,b,c\in \left[ -1;3 \right] \\
& \Rightarrow {{\left( m+g\left( a \right)+g\left( b \right)-g\left( c \right) \right)}^{2}}-2g\left( a \right)g\left( b \right)+2g\left( a \right)g\left( c \right)+2g\left( b \right)g\left( c \right)-2{{g}^{2}}\left( c \right)>0,\forall a,b,c\in \left[ -1;3 \right] \\
& \Rightarrow {{\left( m+g\left( a \right)+g\left( b \right)-g\left( c \right) \right)}^{2}}>2\left( g\left( a \right)-g\left( c \right) \right)\left( g\left( b \right)-g\left( c \right) \right),\forall a,b,c\in \left[ -1;3 \right] \\
& \Rightarrow {{\left( m+g\left( a \right)+g\left( b \right)-g\left( c \right) \right)}^{2}}>2{{\left( \underset{\left[ -1;3 \right]}{\mathop{\max }} g\left( x \right)-\underset{\left[ -1;3 \right]}{\mathop{\min }} g\left( x \right) \right)}^{2}},\forall a,b,c\in \left[ -1;3 \right] \\
& m>\underset{\left[ -1;3 \right]}{\mathop{\max }} g\left( x \right)+20\sqrt{2}-2\underset{\left[ -1;3 \right]}{\mathop{\min }} g\left( x \right)\Rightarrow m\ge 49. \\
\end{aligned}$
Ta có: $\begin{aligned}
& f\left( a \right)+f\left( b \right)>f\left( c \right),\forall a,b,c\in \left[ -1;3 \right]\Rightarrow m>g\left( c \right)-\left( g\left( a \right)+g\left( b \right) \right),\forall a,b,c\in \left[ -1;3 \right] \\
& \Rightarrow m>\underset{\left[ -1;3 \right]}{\mathop{\max }} g\left( x \right)-2\underset{\left[ -1;3 \right]}{\mathop{\min }} g\left( x \right)\Rightarrow m>20. \\
\end{aligned}$
$\begin{aligned}
& {{f}^{2}}\left( a \right)+{{f}^{2}}\left( b \right)>{{f}^{2}}\left( c \right),\forall a,b,c\in \left[ -1;3 \right]\Rightarrow {{\left( m+g\left( a \right) \right)}^{2}}+{{\left( m+g\left( b \right) \right)}^{2}}>{{\left( m+g\left( c \right) \right)}^{2}},\forall a,b,c\in \left[ -1;3 \right] \\
& \Rightarrow {{m}^{2}}+2\left( g\left( a \right)+g\left( b \right)-g\left( c \right) \right)m+{{g}^{2}}\left( a \right)+{{g}^{2}}\left( b \right)-{{g}^{2}}\left( c \right)>0,\forall a,b,c\in \left[ -1;3 \right] \\
& \Rightarrow {{\left( m+g\left( a \right)+g\left( b \right)-g\left( c \right) \right)}^{2}}-2g\left( a \right)g\left( b \right)+2g\left( a \right)g\left( c \right)+2g\left( b \right)g\left( c \right)-2{{g}^{2}}\left( c \right)>0,\forall a,b,c\in \left[ -1;3 \right] \\
& \Rightarrow {{\left( m+g\left( a \right)+g\left( b \right)-g\left( c \right) \right)}^{2}}>2\left( g\left( a \right)-g\left( c \right) \right)\left( g\left( b \right)-g\left( c \right) \right),\forall a,b,c\in \left[ -1;3 \right] \\
& \Rightarrow {{\left( m+g\left( a \right)+g\left( b \right)-g\left( c \right) \right)}^{2}}>2{{\left( \underset{\left[ -1;3 \right]}{\mathop{\max }} g\left( x \right)-\underset{\left[ -1;3 \right]}{\mathop{\min }} g\left( x \right) \right)}^{2}},\forall a,b,c\in \left[ -1;3 \right] \\
& m>\underset{\left[ -1;3 \right]}{\mathop{\max }} g\left( x \right)+20\sqrt{2}-2\underset{\left[ -1;3 \right]}{\mathop{\min }} g\left( x \right)\Rightarrow m\ge 49. \\
\end{aligned}$
Đáp án C.