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14.
\(y'=2x+\dfrac{16}{x^2}=0\Rightarrow\dfrac{2x^3+16}{x}=0\)
\(\Rightarrow x^3=-8\Rightarrow x=-2\)
Ta có:
\(f\left(-4\right)=20\) ; \(f\left(-2\right)=12\) ; \(f\left(-1\right)=17\)
\(\Rightarrow M=20\) ; \(m=12\)
\(\Rightarrow M+m=32\)
15.
\(y'=5x^4-20x^3+15x^2=5x^2\left(x^2-4x+3\right)\)
\(y'=0\Rightarrow x^2-4x+3=0\) (do \(x^2\) cho nghiệm kép nên ko là cực trị \(\Rightarrow\) ko cần xét)
\(\Rightarrow\left[{}\begin{matrix}x=1\\x=3\notin\left[-1;2\right]\end{matrix}\right.\)
\(f\left(-1\right)=-10\) ; \(f\left(1\right)=2\) ; \(f\left(2\right)=-7\)
\(\Rightarrow M=2\in\left(1;4\right)\)
16.
\(y'=6x^2+6x-12=0\Rightarrow\left[{}\begin{matrix}x=1\\x=-2\notin\left[-1;3\right]\end{matrix}\right.\)
\(f\left(-1\right)=14\) ; \(f\left(1\right)=-6\) ; \(f\left(3\right)=46\)
\(\Rightarrow M=46\) ; \(m=-6\)
\(\Rightarrow M+m=40\in\left(39;42\right)\)
17.
Đặt \(f\left(x\right)=\dfrac{x^2+3x+3}{x+1}\)
BPT nghiệm đúng với mọi x thuộc khoảng đã cho khi \(m\le\min\limits_{\left[0;1\right]}f\left(x\right)\)
\(f'\left(x\right)=\dfrac{x^2+2x}{\left(x+1\right)^2}\ge0;\forall x\in\left[0;1\right]\)
\(\Rightarrow f\left(x\right)\) đồng biến trên \(\left[0;1\right]\)
\(\Rightarrow\min\limits_{\left[0;1\right]}f\left(x\right)=f\left(0\right)=3\)
\(\Rightarrow m\le3\)
18.
Đặt \(f\left(x\right)=\dfrac{x^2+3x+3}{x+1}\)
BPT nghiệm đúng với mọi x thuộc đoạn đã cho khi: \(m\ge\max\limits_{\left[0;1\right]}f\left(x\right)\)
Như câu 17, ta thấy \(f\left(x\right)\) đồng biến trên \(\left[0;1\right]\)
\(\Rightarrow\max\limits_{\left[0;1\right]}f\left(x\right)=f\left(1\right)=\dfrac{7}{2}\)
\(\Rightarrow m\ge\dfrac{7}{2}\)
19.
\(y=5x+\dfrac{5}{x}\ge2\sqrt{5x.\dfrac{5}{x}}=10\)
\(\Rightarrow m=10\in D\)
20.
ĐKXĐ: \(-2\le x\le2\)
\(f'\left(x\right)=-\dfrac{x}{\sqrt{4-x^2}}=0\Rightarrow x=0\)
\(f\left(-2\right)=-4\) ; \(f\left(0\right)=-2\) ; \(f\left(2\right)=-4\)
\(\Rightarrow M=-2\) ; \(m=-4\)
\(\Rightarrow M^2+m=0\)
21.
ĐKXĐ: \(-2\le x\le2\)
\(y'=1+\dfrac{x}{\sqrt{4-x^2}}=0\Rightarrow\sqrt{4-x^2}=-x\left(x\le0\right)\)
\(\Rightarrow4-x^2=x^2\)
\(\Rightarrow x^2=2\Rightarrow\left[{}\begin{matrix}x=\sqrt{2}>0\left(loại\right)\\x=-\sqrt{2}\end{matrix}\right.\)
\(f\left(-2\right)=-2\) ; \(f\left(-\sqrt{2}\right)=-2\sqrt{2}\) ; \(f\left(2\right)=2\)
\(\Rightarrow M=2\) ; \(m=-2\sqrt{2}\)
\(\Rightarrow M+m=2-2\sqrt{2}\)
22.
\(f'\left(x\right)=\dfrac{x^2-1}{x^2}=0\Rightarrow\left[{}\begin{matrix}x=-1\notin\left[\dfrac{1}{2};3\right]\\x=1\end{matrix}\right.\)
\(f\left(\dfrac{1}{2}\right)=-\dfrac{5}{2}\) ; \(f\left(1\right)=-3\) ; \(f\left(3\right)=-\dfrac{5}{3}\)
\(\Rightarrow M=-\dfrac{5}{3}\) ; \(m=-3\)
23.
\(f'\left(x\right)=1-\dfrac{225}{x^2}=0\Rightarrow x^2=225\)
\(\Rightarrow\left[{}\begin{matrix}x=15>0\left(loại\right)\\x=-15\end{matrix}\right.\)
\(f\left(-15\right)=-30\)
\(\Rightarrow M=-30\in A\)
