tìm m để y= |-$x^{4}$+8 $x^{2}$ +m| thuộc [-1;3] bằng có Min bằng 2018

Đáp án và giải thích các bước giải:

Ảnh 

Đáp án:

 `m∈{-2034;2027}`

Giải thích các bước giải:

`y=|-x^4+8x^2+m|`

`\text(TXĐ: D=R)`

`\text( Đặt f(x))=-x^4+8x^2+m`

`f'(x)=-4x^3+16x`

`f'(x)=0<=>-4x^3+16x=0`

`<=>-4x.(x^2-4)=0`

`<=>`\(\left[ \begin{array}{l}x=0\\x^2-4=0\end{array} \right.\) 

`<=>`\(\left[ \begin{array}{l}x=0\\x=-2\\x=2\end{array} \right.\) 

`\text( Bảng biến thiên:)`

\begin{array}{|c|cc|}\hline \text{$x$}&\text{$-\infty$}&\text{}&\text{-2}&\text{}&\text{-1}&\text{}&\text{0}&\text{}&\text{}&\text{2}&\text{}&\text{3}&\text{}&\text{$+\infty$}\\\hline \text{$f'(x)$}&\text{}&\text{//}&\text{//}&\text{//}&\text{|}&\text{$-$}&\text{0}&\text{}&\text{+}&\text{0}&\text{$-$}&\text{|}&\text{//}&\text{//}\\\hline \text{$f(x)$}&\text{}&\text{}&\text{}&\text{}&\text{m+7}&\text{}&\text{}&\text{}&\text{}&\text{m+16}\\&\text{}&\text{//}&\text{//}&\text{//}&\text{|}&\text{$\searrow$}&\text{}&\text{}&\text{$\nearrow$}&\text{}&\text{$\searrow$}&\text{|}&\text{//}&\text{//}\\&\text{}&\text{}&\text{}&\text{}&\text{}&\text{}&\text{m}&\text{}&\text{}&\text{}&\text{}&\text{ m-9}\\\hline \end{array}

`->max_{[-1;3]}f(x)=f(2)=m+16;min_{[-1;3]}f(x)=f(3)=m-9`

*TH1: `m-9>0<=>m>9`

`->min_{[-1;3]}y=m-9`

`->m-9=2018`

`->m=2027(tmđk)`

*TH2: `m+16<0<=>m<-16`

`->min_{[-1;3]}y=-(m+16)`

`->-(m+16)=2018`

`->-m-16=2018`

`->m=-2034(tmđk)`

`text( Vậy)m∈{-2034;2027}`