Answer:
Critical value f(1)=2.
Minimum at (1,2), function is decreasing for
and increasing for ![x>1.](https://tex.z-dn.net/?f=x%3E1.)
is point of inflection.
When 0<x<3, function is concave upwards and when x>3, , function is concave downwards.
Step-by-step explanation:
1. Find the domain of the function f(x):
![\left\{\begin{array}{l}x\ge 0\\x\neq 0\end{array}\right.\Rightarrow x>0.](https://tex.z-dn.net/?f=%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bl%7Dx%5Cge%200%5C%5Cx%5Cneq%200%5Cend%7Barray%7D%5Cright.%5CRightarrow%20x%3E0.)
2. Find the derivative f'(x):
![f'(x)=\dfrac{(x+1)'\cdot \sqrt{x}-(x+1)\cdot (\sqrt{x})'}{(\sqrt{x})^2}=\dfrac{\sqrt{x}-\frac{x+1}{2\sqrt{x}}}{x}=\dfrac{2x-x-1}{2x\sqrt{x}}=\dfrac{x-1}{2x^{\frac{3}{2}}}.](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cdfrac%7B%28x%2B1%29%27%5Ccdot%20%5Csqrt%7Bx%7D-%28x%2B1%29%5Ccdot%20%28%5Csqrt%7Bx%7D%29%27%7D%7B%28%5Csqrt%7Bx%7D%29%5E2%7D%3D%5Cdfrac%7B%5Csqrt%7Bx%7D-%5Cfrac%7Bx%2B1%7D%7B2%5Csqrt%7Bx%7D%7D%7D%7Bx%7D%3D%5Cdfrac%7B2x-x-1%7D%7B2x%5Csqrt%7Bx%7D%7D%3D%5Cdfrac%7Bx-1%7D%7B2x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D.)
This derivative is equal to 0 at x=1 and is not defined at x=0. Since x=0 is not a point from the domain, the crititcal point is only x=1. The critical value is
![f(1)=\dfrac{1+1}{\sqrt{1}}=2.](https://tex.z-dn.net/?f=f%281%29%3D%5Cdfrac%7B1%2B1%7D%7B%5Csqrt%7B1%7D%7D%3D2.)
2. For
the derivative f'(x)<0, then the function is decreasing. For
the derivative f'(x)>0, then the function is increasing. This means that point x=1 is point of minimum.
3. Find f''(x):
![f''(x)=\dfrac{(x-1)'\cdot 2x^{\frac{3}{2}}-(x-1)\cdot (2x^{\frac{3}{2}})'}{(2x^{\frac{3}{2}})^2}=](https://tex.z-dn.net/?f=f%27%27%28x%29%3D%5Cdfrac%7B%28x-1%29%27%5Ccdot%202x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D-%28x-1%29%5Ccdot%20%282x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%29%27%7D%7B%282x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%29%5E2%7D%3D)
![=\dfrac{2x^{\frac{3}{2}}-2(x-1)\cdot \frac{3}{2}x^{\frac{1}{2}}}{4x^3}=\dfrac{2x^{\frac{3}{2}}-2\cdot\frac{3}{2}x^{\frac{3}{2}}+ 2\cdot\frac{3}{2}x^{\frac{1}{2}}}{4x^3}=](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B2x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D-2%28x-1%29%5Ccdot%20%5Cfrac%7B3%7D%7B2%7Dx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%7B4x%5E3%7D%3D%5Cdfrac%7B2x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D-2%5Ccdot%5Cfrac%7B3%7D%7B2%7Dx%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%2B%202%5Ccdot%5Cfrac%7B3%7D%7B2%7Dx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%7B4x%5E3%7D%3D)
![=\dfrac{-x+3}{4x^{\frac{5}{2}}}.](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B-x%2B3%7D%7B4x%5E%7B%5Cfrac%7B5%7D%7B2%7D%7D%7D.)
When f''(x)=0, x=3 and
When 0<x<3, f''(x)>0 - function is concave upwards and when x>3, f''(x)>0 - function is concave downwards.
Point
is point of inflection.
THE CORRECT ANSWER IS THE LAST ONE sorry caps :) hops this help tho plzz mark as best plzzzzzzzzzzzzzzz
Answer:
what is the question
Step-by-step explanation:
Answer:
Im pretty sure the answer is B, forgive me if I am wrong.
Step-by-step explanation: