`f(sinx)=3sinx+m`
Đặt `t=sinx`
\begin{array}{|c|cc|}\hline \text{$x$}&\text{$0$}&\text{}&\text{$\dfrac{\pi}{2}$}&\text{}&\text{$\pi$}\\\hline \text{$t=sinx$}&\text{}&\text{}&\text{1}\\&\text{}&\text{$\nearrow$}&\text{}&\text{$\searrow$}\\&\text{0}&\text{}&\text{}&\text{}&\text{0}\\\hline\end{array}
`=>t∈(0;1]`
Xét :`f(t)=3t+m;t∈(0;1]`
`=>m=f(t)-3t(**)`
Xét `g(t)=f(t)-3t`
`g'(t)=f'(t)-3`
`∀t∈(0;1]` hàm số `f(t)` nghịch biến trên `(0;1]`
`=>f'(t)<0;∀t∈(0;1]=>g'(t)<0;∀t∈(0;1]`
Bảng biến thiên:
\begin{array}{|c|cc|}\hline \text{$t$}&\text{0}&\text{}&\text{1}\\\hline \text{$g'(t)$}&\text{}&\text{$-$}&\text{}\\\hline \text{$g(t)$}&\text{1}&\text{}&\text{}\\&\text{}&\text{$\searrow$}&\text{}\\&\text{}&\text{}&\text{-4}\\\hline \end{array}
$\buildrel{{\text{Theo đồ thị f(t)}}}\over\longrightarrow f(1)=-1;f(0)=1$
`g(0)=f(0)-3.0=1;g(1)=f(1)-3=-1-3=-4`
Để phương trình `(**)` có nghiệm thì `-4<=m<1`
Vậy `m∈[-4;1)`
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Cách `2` : Ghép trục.
