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Giải như sau.
(1)+(2)⇔x2−2x+1+√x2−2x+5=y2+√y2+4⇔(x2−2x+5)+√x2−2x+5=y2+4+√y2+4⇔√y2+4=√x2−2x+5⇒x=3y(1)+(2)⇔x2−2x+1+x2−2x+5=y2+y2+4⇔(x2−2x+5)+x2−2x+5=y2+4+y2+4⇔y2+4=x2−2x+5⇒x=3y
⇔√y2+4=√x2−2x+5⇔y2+4=x2−2x+5, chỗ này do hàm số f(x)=t2+tf(x)=t2+t đồng biến ∀t≥0∀t≥0
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a) \(f\left(x\right)=5x^3-7x^2+2x+5\)
\(\Rightarrow f\left(1\right)=5.1^3-7.1^2+2.1+5\)
\(\Rightarrow f\left(1\right)=5.1-7.1+2+5\)
\(\Rightarrow f\left(1\right)=5-7+7\)
\(\Rightarrow f\left(1\right)=5\)
Vậy f(1) = 5.
\(g\left(x\right)=7x^3-7x^2+2x+5\)
\(\Rightarrow g\left(\frac{1}{2}\right)=7.\left(\frac{1}{2}\right)^3-7.\left(\frac{1}{2}\right)^2+2.\frac{1}{2}+5\)
\(\Leftrightarrow g\left(\frac{1}{2}\right)=7.\frac{1}{8}-7.\frac{1}{4}+1+5\)
\(\Leftrightarrow g\left(\frac{1}{2}\right)=\frac{7}{8}-\frac{14}{8}+6\)
\(\Leftrightarrow g\left(\frac{1}{2}\right)=\frac{-7}{8}+\frac{48}{8}\)
\(\Leftrightarrow g\left(\frac{1}{2}\right)=\frac{41}{8}\)
Vậy \(g\left(\frac{1}{2}\right)=\frac{41}{8}\)
\(h\left(x\right)=2x^3+4x+1\)
\(\Rightarrow h\left(0\right)=2.0^3+4.0+1\)
\(\Rightarrow h\left(0\right)=0+0+1\)
\(\Rightarrow h\left(0\right)=1\)
Vậy \(h\left(0\right)=1\)
Lời giải:
Ta có:
\(h(x)=f(x)-g(x)=(-5x^5-x^5+2x^4-x^2-1)-(-6+2x-2x^3-x^4+3x^5)\)
\(=(-5x^5-x^5-3x^5)+(2x^4+x^4)+2x^3-x^2-2x+(-1+6)\)
\(=-9x^5+3x^4+2x^3-x^2-2x+5\)
\(\Rightarrow \left\{\begin{matrix} h(-1)=-9(-1)^5+3(-1)^4+2(-1)^3-(-1)^2-2(-1)+5=16\\ h(1)=-9.1^5+3.1^4+2.1^3-1^2-2.1+5=-2\\ h(-2)=-9(-2)^5+3(-2)^4+2(-2)^3-(-2)^2-2(-2)+5=325\\ h(2)=-9.2^5+3.2^4+2.2^3-2^2-2.2+5=-227\end{matrix}\right.\)
\(q(x)=g(x)-f(x)=-[f(x)-g(x)]=-h(x)\)
\(\Rightarrow q(-1)=-h(-1)=-16\)
\(q(1)=-h(1)=2\)
\(q(-2)=-h(-2)=-325\)
\(q(2)=-h(2)=227\)
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a. f(x)+g(x)=2x5−4x4+3x3−x2+5x−1+(−x5+2x4−3x3−x2−2x+7)
=2x5-x5-4x4+2x4+3x3-3x3-x2-x2+5x-2x-1+7
=x5-2x4-2x2+3x+6
b. f(x)+h(x)=2x5−4x4+3x3−x2+5x−1+x5−2x4−2x2−x−3
=2x5+x5-4x4-2x4+3x3-x2-2x2+5x-x-1-3
=3x5-6x4+3x3-3x2+6x-4
c. g(x)+h(x)=−x5+2x4−3x3−x2−2x+7+x5−2x4−2x2−x−3
=-x5+x5+2x4-2x4-3x3-x2-2x2-2x-x+7-3
=-3x3-3x2-3x+4
d. f(x)-g(x)=2x5−4x4+3x3−x2+5x−1-(−x5+2x4−3x3−x2−2x+7)
=2x5−4x4+3x3−x2+5x−1-x5-2x4+3x3+x2+2x-7
=2x5-x5-4x4-2x4+3x3+3x3-x2+x2+5x+2x-1-7
=x5-6x4+6x3+7x-8
e. f(x)-h(x)=2x5−4x4+3x3−x2+5x−1-(x5−2x4−2x2−x−3)
=2x5−4x4+3x3−x2+5x−1-x5+2x4+2x2+x+3
=2x5-x5-4x4+2x4+3x3-x2+2x2+5x+x-1+3
=x5-2x4+3x3+x2+6x-4
h. g(x)-h(x)=−x5+2x4−3x3−x2−2x+7-(x5−2x4−2x2−x−3)
=−x5+2x4−3x3−x2−2x+7-x5+2x4+2x2+x+3
=-x5-x5+2x4+2x4-3x3-x2+2x2-2x+x+7+3
=-2x5+4x4-3x3+x2-x+10
f. f(x)+g(x)+h(x)=2x5−4x4+3x3−x2+5x−1+(−x5+2x4−3x3−x2−2x+7)+x5−2x4−2x2−x−3
=2x5-x5+x5-4x4+2x4-2x4+3x3-3x3-x2-x2-2x2+5x-2x-x-1+7-3
=2x5-4x4-4x2+2x+3
g. f(x)+g(x)-h(x)=2x5−4x4+3x3−x2+5x−1+(−x5+2x4−3x3−x2−2x+7)-(x5−2x4−2x2−x−3)
=2x5−4x4+3x3−x2+5x−1+(−x5+2x4−3x3−x2−2x+7)-x5+2x4+2x2+x+3
=2x5-x5-x5-4x4+2x4+2x4+3x3-3x3-x2-x2+2x2+5x-2x+x-1+7+3
=4x+9
n. f(x)-g(x)+h(x)=2x5−4x4+3x3−x2+5x−1-(−x5+2x4−3x3−x2−2x+7)+x5−2x4−2x2−x−3
=2x5−4x4+3x3−x2+5x−1-x5-2x4+3x3+x2+2x-7+x5−2x4−2x2−x−3
=2x5-x5+x5-4x4-2x4-2x4+3x3+3x3-x2+x2-2x2+5x+2x-x-1-7-3
=2x5-8x4+6x3-2x2+6x-11
m. f(x)-g(x)-h(x)=2x5−4x4+3x3−x2+5x−1-(−x5+2x4−3x3−x2−2x+7)-(x5−2x4−2x2−x−3)
=2x5−4x4+3x3−x2+5x−1-x5-2x4+3x3+x2+2x-7-x5+2x4+2x2+x+3
=2x5-x5-x5-4x4-2x4+2x4+3x3+3x3-x2+x2+2x2+5x+2x+x-1-7+3
=-4x4+6x3+2x2+8x-5