18 is not a function because a function is when exactly ONE x value is paired with ONE y value.
19 is not a function because of the same logic for number 18
Answer:
4
Step-by-step explanation:
![slope = \frac{5 - 1}{5 - 4} \\ = \frac{4}{1} \\ = 4](https://tex.z-dn.net/?f=slope%20%3D%20%20%5Cfrac%7B5%20-%201%7D%7B5%20-%204%7D%20%20%5C%5C%20%20%3D%20%20%5Cfrac%7B4%7D%7B1%7D%20%20%20%5C%5C%20%20%3D%204)
Answer:
a)![X=((-15-\sqrt{201},(-15+\sqrt{201}),(0,\infty)](https://tex.z-dn.net/?f=X%3D%28%28-15-%5Csqrt%7B201%7D%2C%28-15%2B%5Csqrt%7B201%7D%29%2C%280%2C%5Cinfty%29)
b)![Y=(\infty,\frac{1}{2}(-15-\sqrt{201} ) ),(\frac{1}{2}()-15+\sqrt{201)},0 )](https://tex.z-dn.net/?f=Y%3D%28%5Cinfty%2C%5Cfrac%7B1%7D%7B2%7D%28-15-%5Csqrt%7B201%7D%20%29%20%29%2C%28%5Cfrac%7B1%7D%7B2%7D%28%29-15%2B%5Csqrt%7B201%29%7D%2C0%20%20%29)
Step-by-step explanation:
From the question we are told that
The Function
![f(x)=1+\frac{1}{x} +\frac{5}{x^2} +\frac{1}{x^3}](https://tex.z-dn.net/?f=f%28x%29%3D1%2B%5Cfrac%7B1%7D%7Bx%7D%20%20%2B%5Cfrac%7B5%7D%7Bx%5E2%7D%20%2B%5Cfrac%7B1%7D%7Bx%5E3%7D)
Generally the differentiation of function f(x) is mathematically solved as
![f(x)=1+\frac{1}{x} +\frac{5}{x^2} +\frac{1}{x^3}](https://tex.z-dn.net/?f=f%28x%29%3D1%2B%5Cfrac%7B1%7D%7Bx%7D%20%20%2B%5Cfrac%7B5%7D%7Bx%5E2%7D%20%2B%5Cfrac%7B1%7D%7Bx%5E3%7D)
![f(x)=\frac{x^3+x^2+5x+1}{x^2}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7Bx%5E3%2Bx%5E2%2B5x%2B1%7D%7Bx%5E2%7D)
Therefore
![f'(x)=\frac{x^2+10x+3}{x^4}](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cfrac%7Bx%5E2%2B10x%2B3%7D%7Bx%5E4%7D)
Generally critical point is given as
![f'(x)=0](https://tex.z-dn.net/?f=f%27%28x%29%3D0)
![\frac{x^2+10x+3}{x^4}=0](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E2%2B10x%2B3%7D%7Bx%5E4%7D%3D0)
![x=-5 \pm\sqrt{22}](https://tex.z-dn.net/?f=x%3D-5%20%5Cpm%5Csqrt%7B22%7D)
Generally the maximum and minimum x value for critical point is mathematically solved as
![f'(-5 \pm\sqrt{22})](https://tex.z-dn.net/?f=f%27%28-5%20%5Cpm%5Csqrt%7B22%7D%29)
Where
Maximum value of x
![f'(-5 +\sqrt{22})](https://tex.z-dn.net/?f=f%27%28-5%20%2B%5Csqrt%7B22%7D%29%3C0)
Minimum value of x
![f'(-5 +\sqrt{22})](https://tex.z-dn.net/?f=f%27%28-5%20%2B%5Csqrt%7B22%7D%29%3C0)
Therefore interval of increase is mathematically given by
![f'(-5 -\sqrt{22}),f'(-5 +\sqrt{22})](https://tex.z-dn.net/?f=f%27%28-5%20-%5Csqrt%7B22%7D%29%2Cf%27%28-5%20%2B%5Csqrt%7B22%7D%29)
![f(x)](https://tex.z-dn.net/?f=f%28x%29%3C0%2C-%5Cinfty%3Cx%3C%28f%27%28-5%20-%5Csqrt%7B22%7D%29%29%20%2C%28f%27%28-5%20%2B%5Csqrt%7B22%7D%29%29%3Cx%3C0%2C0%3Cx%3C%20%5Cinfty)
Therefore interval of decrease is mathematically given by
![(-\infty,-5 -\sqrt{22}),f'(-5 +\sqrt{22},0),(0,\infty)](https://tex.z-dn.net/?f=%28-%5Cinfty%2C-5%20-%5Csqrt%7B22%7D%29%2Cf%27%28-5%20%2B%5Csqrt%7B22%7D%2C0%29%2C%280%2C%5Cinfty%29)
Generally the second differentiation of function f(x) is mathematically solved as
![f''(x)=\frac{2(x^2+15x+6)}{x^5}](https://tex.z-dn.net/?f=f%27%27%28x%29%3D%5Cfrac%7B2%28x%5E2%2B15x%2B6%29%7D%7Bx%5E5%7D)
Generally the point of inflection is mathematically solved as
![f''(x)=0](https://tex.z-dn.net/?f=f%27%27%28x%29%3D0)
![x^2+15x+6=0](https://tex.z-dn.net/?f=x%5E2%2B15x%2B6%3D0)
Therefore inflection points is given as
![x=\frac{1}{2} (-15 \pm \sqrt{201}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B1%7D%7B2%7D%20%28-15%20%5Cpm%20%5Csqrt%7B201%7D)
![f''(x)>0,\frac{1}{2}(-15-\sqrt{201})](https://tex.z-dn.net/?f=f%27%27%28x%29%3E0%2C%5Cfrac%7B1%7D%7B2%7D%28-15-%5Csqrt%7B201%7D%29%20%3Cx%3C%5Cfrac%7B1%7D%7B2%7D%28-15-%5Csqrt%7B201%7D%29%20%3Cx%3C0)
a)Generally the concave upward interval X is mathematically given as
![X=((-15-\sqrt{201},(-15+\sqrt{201}),(0,\infty)](https://tex.z-dn.net/?f=X%3D%28%28-15-%5Csqrt%7B201%7D%2C%28-15%2B%5Csqrt%7B201%7D%29%2C%280%2C%5Cinfty%29)
![f''(x)](https://tex.z-dn.net/?f=f%27%27%28x%29%3C0%2C-%5Cinfty%3Cx%3C%5Cfrac%7B1%7D%7B2%7D%28-15-%5Csqrt%7B201%7D%29%20%2C%20%5Cfrac%7B1%7D%7B2%7D%28-15-%5Csqrt%7B201%7D%29%20%3Cx%3C0)
b)Generally the concave downward interval Y is mathematically given as
![Y=(\infty,\frac{1}{2}(-15-\sqrt{201} ) ),(\frac{1}{2}()-15+\sqrt{201)},0 )](https://tex.z-dn.net/?f=Y%3D%28%5Cinfty%2C%5Cfrac%7B1%7D%7B2%7D%28-15-%5Csqrt%7B201%7D%20%29%20%29%2C%28%5Cfrac%7B1%7D%7B2%7D%28%29-15%2B%5Csqrt%7B201%29%7D%2C0%20%20%29)