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.
Answer:
y= -2x -8
Step-by-step explanation:
I will be writing the equation of the perpendicular bisector in the slope-intercept form which is y=mx +c, where m is the gradient and c is the y-intercept.
A perpendicular bisector is a line that cuts through the other line perpendicularly (at 90°) and into 2 equal parts (and thus passes through the midpoint of the line).
Let's find the gradient of the given line.
![\boxed{gradient = \frac{y1 -y 2}{x1 - x2} }](https://tex.z-dn.net/?f=%5Cboxed%7Bgradient%20%3D%20%20%5Cfrac%7By1%20-y%202%7D%7Bx1%20-%20x2%7D%20%7D)
Gradient of given line
![= \frac{1 - ( - 5)}{3 - ( - 9)}](https://tex.z-dn.net/?f=%20%3D%20%20%5Cfrac%7B1%20-%20%28%20-%205%29%7D%7B3%20-%20%28%20-%209%29%7D%20)
![= \frac{1 + 5}{3 + 9}](https://tex.z-dn.net/?f=%20%3D%20%20%5Cfrac%7B1%20%2B%205%7D%7B3%20%2B%209%7D%20)
![= \frac{6}{12}](https://tex.z-dn.net/?f=%20%3D%20%20%5Cfrac%7B6%7D%7B12%7D%20)
![= \frac{1}{2}](https://tex.z-dn.net/?f=%20%3D%20%20%20%5Cfrac%7B1%7D%7B2%7D%20)
The product of the gradients of 2 perpendicular lines is -1.
(½)(gradient of perpendicular bisector)= -1
Gradient of perpendicular bisector
= -1 ÷(½)
= -1(2)
= -2
Substitute m= -2 into the equation:
y= -2x +c
To find the value of c, we need to substitute a pair of coordinates that the line passes through into the equation. Since the perpendicular bisector passes through the midpoint of the given line, let's find the coordinates of the midpoint.
![\boxed{midpoint = ( \frac{x1 + x2}{2} , \frac{y1 + y2}{2}) }](https://tex.z-dn.net/?f=%5Cboxed%7Bmidpoint%20%3D%20%28%20%5Cfrac%7Bx1%20%2B%20x2%7D%7B2%7D%20%2C%20%5Cfrac%7By1%20%2B%20y2%7D%7B2%7D%29%20%20%7D)
Midpoint of given line
![= ( \frac{3 - 9}{2} , \frac{1 - 5}{2} )](https://tex.z-dn.net/?f=%20%3D%20%28%20%5Cfrac%7B3%20%20-%20%209%7D%7B2%7D%20%2C%20%5Cfrac%7B1%20-%205%7D%7B2%7D%20%29)
![= ( \frac{ - 6}{2} , \frac{ - 4}{2} )](https://tex.z-dn.net/?f=%20%3D%20%28%20%5Cfrac%7B%20-%206%7D%7B2%7D%20%20%2C%20%5Cfrac%7B%20-%204%7D%7B2%7D%20%29)
![= ( - 3 , - 2)](https://tex.z-dn.net/?f=%20%3D%20%28%20-%203%20%2C%20-%202%29)
Substituting (-3, -2) into the equation:
-2= -2(-3) +c
-2= 6 +c
c= -2 -6 <em>(</em><em>-</em><em>6</em><em> </em><em>on both</em><em> </em><em>sides</em><em>)</em>
c= -8
Thus, the equation of the perpendicular bisector is y= -2x -8.
Answer:
268
Step-by-step explanation:
add 126 then subtract 308
Answer:
Mr. Holms used 1/10 of the whole carton for each serving.
Step-by-step explanation:
Mr. Holms used 4/5 of the carton, which means he has 1/5 of the carton remaining from a whole carton and if he divides 1/5 among 2 servings he gets 1/5 divided by 2 is 1/10.
Check Your Answer: 4/5 divided among 2 servings would be 4/5+ 1/10 + 1/10 = 8/10 + 1/10 + 1/10 = 10/10 = 1 whole carton.
Answer:
4??
Step-by-step explanation:
Although, philosophers would love to give you a different answer (like 5)