1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
fomenos
3 years ago
14

Find the x-coordinates of any relative extrema and inflection point(s) for the function f(x)= 6x^1/3 + 3x^4/3. You must justify

your answer using an analysis of f'(x) and f''(x)
My work so far
F'(x) =2/(x^2/3) + 4x^1/3
​
Mathematics
1 answer:
irina [24]3 years ago
7 0

Answer:

x-coordinates of relative extrema = \frac{-1}{2}

x-coordinates of the inflexion points are 0, 1

Step-by-step explanation:

f(x)=6x^{\frac{1}{3}}+3x^{\frac{4}{3}}

Differentiate with respect to x

f'(x)=6\left ( \frac{1}{3} \right )x^{\frac{-2}{3}}+3\left ( \frac{4}{3} \right )x^{\frac{1}{3}}=\frac{2}{x^{\frac{2}{3}}}+4x^{\frac{1}{3}}

f'(x)=0\Rightarrow \frac{2}{x^{\frac{2}{3}}}+4x^{\frac{1}{3}}=0\Rightarrow x=\frac{-1}{2}

Differentiate f'(x) with respect to x

f''(x)=2\left ( \frac{-2}{3} \right )x^{\frac{-5}{3}}+\frac{4}{3}x^{\frac{-2}{3}}=\frac{-4x^{\frac{2}{3}}+4x^{\frac{5}{3}}}{3x^{\frac{2}{3}}x^{\frac{5}{3}}}\\f''(x)=0\Rightarrow \frac{-4x^{\frac{2}{3}}+4x^{\frac{5}{3}}}{3x^{\frac{2}{3}}x^{\frac{5}{3}}}=0\Rightarrow x=1

At x = \frac{-1}{2},

f''\left ( \frac{-1}{2} \right )=\frac{4\left ( -1+4\left ( \frac{-1}{2} \right ) \right )}{3\left ( \frac{-1}{2} \right )^{\frac{5}{3}}}>0

We know that if f''(a)>0 then x = a is a point of minima.

So, x=\frac{-1}{2} is a point of minima.

For inflexion points:

Inflexion points are the points at which f''(x) = 0 or f''(x) is not defined.

So, x-coordinates of the inflexion points are 0, 1

You might be interested in
Together, Emily and Charlotte have a total of 60 strawberries. Emily sats 4 times the amount that Charlotte eats. How many straw
ehidna [41]

Answer:

12

Step-by-step explanation:

Since Emily eats four times the amount that Charlotte eats, then Emily will eat: = (4 × x) = 4x. Therefore, Charlotte eats 12 strawberries

7 0
2 years ago
What is the rate of 70 miles in 2 hours in unit rate
tamaranim1 [39]
You want one of the values in the ratio to be one for a unit rate. First, write out the ratio. It is 70 miles/2 hours. It's better not to include decimal numbers when you write a ratio when you can, so we're going to make the hours the unit rate. To make 2 -> 1 hour, divide both sides of the ratio by 2. 70/2 = 35. 
The unit rate is 35 mi./hr.
Hope that helped you.
5 0
3 years ago
Read 2 more answers
Ursula picks carrots and radishes from her garden. She picks
densk [106]
2 1/2 more pounds of carrots.
7 0
3 years ago
Find the value of f(3) for the function.<br> f(x) = -2(x + 3)
BabaBlast [244]

Answer:

-2(3+3) = -12

Step-by-step explanation:

f(x) is telling you to substitute all the x values into the equation. The f(x) of the function is 3, telling you to plug in 3 in the equation. This leads to -2(6) = -12.

6 0
2 years ago
Read 2 more answers
Help me pleaseeeeeee
castortr0y [4]
Use the fomula a_(n)= a_(1)+ d(n-1)
4 0
3 years ago
Other questions:
  • 3220 divided by 3 equals
    8·1 answer
  • Can someone help on this midpoint question please
    10·1 answer
  • The bill (parts and labor) for the repair of a car is $1110.50. The cost of the parts is $575. If the car mechanic chargers $42
    9·1 answer
  • What is the answer now in two minutes
    6·1 answer
  • What is the equation of this line that models the data on the graph?<br> A. y= x/3 + 2<br> B. y=3x+2
    13·1 answer
  • Select one:<br> A. $5,300<br> B. $890<br> C. $12,400<br> D. $13,750
    11·1 answer
  • a bullet fired from a gun traveled 3,600 m south in 6.0 seconds. What was its velocity in (a) m/sec and (b) km/hour?​
    6·1 answer
  • The graph of the function is symmetric with respect to the origin. f(x)=(1/5x)-x^19. T or F
    12·1 answer
  • Which of the following shows the graphic of y=In(-2x)?​
    9·1 answer
  • Colleen buys a movie for $30 and pays 8% sales tax. Her cousin, Brad, lives in another state. Brad buys the same movie for $32 a
    6·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!