Find the slope and the y-intercept of the equation 2x − 3y = 6.

Find the critical numbers of the function f(x) = x6(x − 2)5.x = (smallest value)x = x = (largest value)(b) What does the Second

Marrrta [24]

Answer:

$a) x=0, x=\frac{12}{11}, x=2 \:$ b) The 2nd Derivative test shows us the change of sign and concavity at some point. c) At which point the concavity changes or not. This is only possible with the 2nd derivative test.

Step-by-step explanation:

a) To find the critical numbers, or critical points of:

$f(x)=x^{6}(x-2)^{5}$

1) The procedure is to calculate the 1st derivative of this function. Notice that in this case, we'll apply the <em>Product Rule</em> since there is a product of two functions.

$f(x)=x^{6}(x-2)^{5}\Rightarrow f'(x)=(f*g)'(x)\\=f'g+fg'\Rightarrow (fg)'(x)=6x^{5}(x-2)^{5}+5x^{6}(x-2)^{4} \Rightarrow 6x^{5}(x-2)^{5}+5x^{6}(x-2)^{4}=0\\f'(x)=6x^{5}(x-2)^{5}+5x^{6}(x-2)^{4}$

2) After that, set this an equation then find the values for x.

$x^{5}(x-2)^{4}[6(x-2)+5x]=0\Rightarrow x^{5}(x-2)^{4}[11x-12]=0\Rightarrow x_{1}=0\\(x-2)^{4}=0\Rightarrow \sqrt[4]{(x-2)}=\sqrt[4]{0}\Rightarrow x-2=0\Rightarrow x_{2}=2\\(11x-12)=0\Rightarrow x_{3}=\frac{12}{11}$

$x=0\:(smallest\:value)\:x_{3}=\frac{12}{11}\:x=2 (largest value)$

b) The Second Derivative Test helps us to check the sign of given critical numbers.

Rewriting f'(x) factorizing:

$f'(x)=(11x-12)(x-2)^4x^{5}$

Applying product Rule to find the 2nd Derivative, similarly to 1st derivative:

$f''(x)>0 \Rightarrow Concavity\: Up\\\\f''(x)$

$f''(x)=11\left(x-2\right)^4x^5+4\left(x-2\right)^3x^5\left(11x-12\right)+5\left(x-2\right)^4x^4\left(11x-12\right)\\f''(x)=10\left(x-2\right)^3x^4\left(11x^2-24x+12\right)$

1) Setting this to zero, as an equation:

$10\left(x-2\right)^3x^4\left(11x^2-24x+12\right)=0\\\\$

$10\left(x-2\right)^3x^4\left(11x^2-24x+12\right)=0\\(x-2)^{3}=0 \Rightarrow x_1=2\\x^{4}=0 \therefore x_2=0\\11x^{2}-24x+12=0 \Rightarrow x_3=\frac{12+2\sqrt{3}}{11}\:,x_4=\frac{12-2\sqrt{3}}{11}\cong 0.78$

2) Now, let's define which is the inflection point, the domain is as a polynomial function:

$D=(-\infty$

Looking at the graph.

Plugging these inflection points in the original equation $f(x)=x^{6}(x-2)^{5}$ to get y coordinate:

We have as Inflection Points and their respective y coordinates (Converting to approximate decimal numbers)

$(1.09,-1.05)$ Inflection Point and Local Minimum

$(2,0)$ Inflection Point and Saddle Point

$(0,0)$ Inflection Point Local Maximum

(Check the graph)

c) At which point the concavity changes or not. This is only possible with the 2nd derivative test.

At

$x=\frac{12}{11}\cong1.09$ Local Minimum

$At\:x=0,\:Local \:Maximum$

$At\:x=2, \:neither\:a\:minimum\:nor\:a\:maximum$ (Saddle Point)

5 0

3 years ago

A dresser has a width of 6 feet, a height of 3 feet, and a depth of 2 feet. If the dresser has 6 drawers of the same size, what

jeka57 [31]

Answer:

The volume of the space for each drawer is 6 feet^3.

Step-by-step explanation:

Solve for the volume of the entire dresser.

Formula for volume of a rectangle:

Volume (v) = Length (l) × width (w) × height (h)

v = 2 × 6 × 3

v = 12 × 3

v = 36

Now divide the volume by the number of drawers.

36 ÷ 6 = 6

6 0

3 years ago

Y=.04x^2+8.3x+4.3. How would I solve this?

tiny-mole [99]

It helps if you use the website math way, you punch in your equations and it gives you the answer instantly

6 0

4 years ago

Determine whether the lines are parallel, perpendicular or neither:

11Alexandr11 [23.1K]

Answer:

They are parallel lines

Step-by-step explanation:

Firsr you have to put both of the equations in slope intercept form which is y=mx+b and line 1 that would be 4y=3x+12 then you divide the slope and the y-intercept by 4 to get ride of the 4 in front of the y and it would equal y=3/4x+3 and for line 2 you do the same and that put into slope intercept form would be 2y=1.5x-14 so you do the same and divide the slope and the y-intercept by 2 which would equal y=0.75x-7 and 3/4 in decimal form would be 0.75. since the slopes are the same the lines would be parallel

8 0

4 years ago

Solve this problem using the above process. The sides of a triangular lot are represented by x, 3x, and 3x + 2. Find each side i

stiks02 [169]

We have 3 sides
x,3x,3x+2
P=632
x+3x+3x+2=632
7x=632-2
7x=630
x=90
3x=270
3x+2=272

5 0

3 years ago

Read 2 more answers