(2х²+3х-5)/(х+1)>0
2x²+3x-5=0
D=9+40=46
x1=(-3-7)/4=-2,5 x2=(-3+7)/4=1
x+1=0 x=-1
_ + _ +
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-2,5 -1 1
x∈[-2,5;-1) U [1;≈)
(2х²+3х-5)/(х+1)≤3
(2х²+3х-5-3x-3)/(х+1)≤0
(2х²-8)/(х+1)≤0
2(x-2)(x+2)/(x+1)≤0
x=-2 x=-1 x=2
_ + _ +
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-2 -1 2
x∈(-≈;-2] U (-1;2]
Объединим решения x∈[-2,5;-1) U [1;≈) и x∈(-≈;-2] U (-1;2]⇒х∈[-2,5;-2] U [1;2]
![1)\,\,\, f(x)= \sqrt[3]{x^2+x} ,\,\,\,\, x=1.004](https://tex.z-dn.net/?f=1%29%5C%2C%5C%2C%5C%2C+f%28x%29%3D+%5Csqrt%5B3%5D%7Bx%5E2%2Bx%7D+%2C%5C%2C%5C%2C%5C%2C%5C%2C+x%3D1.004)
Вычислить приближенно будем использовать следующую формулу:
![f(x_0+зx)\approxf(x_0)+d[f(x_0)]](https://tex.z-dn.net/?f=f%28x_0%2B%D0%B7x%29%5Capproxf%28x_0%29%2Bd%5Bf%28x_0%29%5D)
Данном примере
![x_0=1;\,\,\,\, зx=0.004](https://tex.z-dn.net/?f=x_0%3D1%3B%5C%2C%5C%2C%5C%2C%5C%2C+%D0%B7x%3D0.004)
![f(x_0)=f(1)= \sqrt[3]{1^2+1} = \sqrt[3]{2}](https://tex.z-dn.net/?f=f%28x_0%29%3Df%281%29%3D+%5Csqrt%5B3%5D%7B1%5E2%2B1%7D+%3D+%5Csqrt%5B3%5D%7B2%7D+)
![d[f(x_0)]=f'(x_0)зx](https://tex.z-dn.net/?f=d%5Bf%28x_0%29%5D%3Df%27%28x_0%29%D0%B7x)
Вычислим производную функции
![f'(x)=( \sqrt[3]{x^2+x} )'= \frac{1+2x}{3(x^2+x)^{2/3}}](https://tex.z-dn.net/?f=f%27%28x%29%3D%28++%5Csqrt%5B3%5D%7Bx%5E2%2Bx%7D+%29%27%3D+%5Cfrac%7B1%2B2x%7D%7B3%28x%5E2%2Bx%29%5E%7B2%2F3%7D%7D+)
Значение производной в точке х0=1
![f'(1)= \frac{1+2\cdot1}{3(1^2+1)^{2/3}} = \frac{ \sqrt[3]{2} }{2 }](https://tex.z-dn.net/?f=f%27%281%29%3D+%5Cfrac%7B1%2B2%5Ccdot1%7D%7B3%281%5E2%2B1%29%5E%7B2%2F3%7D%7D+%3D+%5Cfrac%7B+%5Csqrt%5B3%5D%7B2%7D+%7D%7B2+%7D+)
`
![d[f(1)]= \frac{ \sqrt[3]{2} }{2} \cdot0.04=0.02 \sqrt[3]{2}](https://tex.z-dn.net/?f=d%5Bf%281%29%5D%3D+%5Cfrac%7B+%5Csqrt%5B3%5D%7B2%7D+%7D%7B2%7D+%5Ccdot0.04%3D0.02+%5Csqrt%5B3%5D%7B2%7D+)
Окончательно имеем, что
![f(1.004)\approx \sqrt[3]{2} -0.02 \sqrt[3]{2} =0.98 \sqrt[3]{2}](https://tex.z-dn.net/?f=f%281.004%29%5Capprox+%5Csqrt%5B3%5D%7B2%7D+-0.02+%5Csqrt%5B3%5D%7B2%7D+%3D0.98+%5Csqrt%5B3%5D%7B2%7D+)
Но это не точно приближенно, может условие я не так переписал.
2)
![f(x)=x^6,\,\,\,\,\,\,\,\, x=2.95](https://tex.z-dn.net/?f=f%28x%29%3Dx%5E6%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C+x%3D2.95)
в данном случае
![x_0=2;\,\,\,\, зx=0.95](https://tex.z-dn.net/?f=x_0%3D2%3B%5C%2C%5C%2C%5C%2C%5C%2C+%D0%B7x%3D0.95)
Найдем значение функции в точке х0
![f(x_0)=f(2)=2^6=64](https://tex.z-dn.net/?f=f%28x_0%29%3Df%282%29%3D2%5E6%3D64)
Вычисляем производную функции
![f'(x)=(x^6)'=6x^5](https://tex.z-dn.net/?f=f%27%28x%29%3D%28x%5E6%29%27%3D6x%5E5)
Найдем значение производной функции в точке х0
![f'(1)=6\cdot 2^5=6\cdot 32=192](https://tex.z-dn.net/?f=f%27%281%29%3D6%5Ccdot+2%5E5%3D6%5Ccdot+32%3D192)
![d[f(2)]=192\cdot0.95 =182.4](https://tex.z-dn.net/?f=d%5Bf%282%29%5D%3D192%5Ccdot0.95+%3D182.4)
Окончательно получаем:
![f(2.95)\approx64+182.4=246.4](https://tex.z-dn.net/?f=f%282.95%29%5Capprox64%2B182.4%3D246.4)