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Andru [333]
2 years ago
9

Find the critical numbers of the function f(x) = x6(x − 1)5 what does the first derivative test tell you that the second derivat

ive test does not?

Mathematics
1 answer:
patriot [66]2 years ago
8 0
The given function is f(x) = x⁶(x-1)⁵

The first derivative is
f'(x) = 6x⁵(x-1)⁵ + 5x⁶(x-1)⁴
       = x⁵(x-1)⁴(6x - 6 + 5x)
       = x⁵(x-1)⁴(11x - 6)
The critical values are the zeros of f'(x). They are
x = 0, 6/11, and 1.
The critical values indicate that turning points exist for f(x) at the critical points. However, we do not know the nature of the turning points.

Write the first derivative in the form f'(x) = (x-1)⁴(11x⁶ - 6x⁵).
The second derivative is
f''(x) = 4(x-1)³(11x⁶ - 6x⁵) + (x-1)⁴(66x⁵ - 30x⁴)
        = (x-1)³(44x⁶ - 24x⁵ + 66x⁶ - 30x⁵ - 66x⁵ + 30x⁴)
        = (x-1)³(110x⁶ - 120x⁵ + 30x⁴)
        = 10x⁴(x-1)³(11x² - 12x + 3)
The sign of f''(x) at the critical values tell us the nature of the turning point.

f''(0) = 0, therefore a point of inflection exists at x = 0.
f''(6/11) > 0, therefore a local minumum exists at x = 6/11.
f''(1) = 0, therefore a point of inflection exists at x=1.

The graphs shown below confirm these results.

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Yuki888 [10]

9514 1404 393

Answer:

  A, C

Step-by-step explanation:

First of all, recognize that any basic arithmetic operation between a rational number and an irrational number will have an irrational result. That means, you can eliminate any answer choices that involve an irrational number.

Fractions and mixed numbers involving integers are <em>always rational</em>, so you only need to look at the radicals. You know 9 is 3×3, so is a perfect square, and √9 = 3 is rational. You know 5 is not a perfect square, so <em>√5 is irrational</em>.

Considering the above, any expression involving √5 will have an irrational result. That means you can eliminate choices B and D.

  choices A and C will result in a rational number

_____

<em>Additional comment</em>

  A = 6.8 +0.75 = 7.55 (rational)

  C = 6.8 × 0.75 = 5.100 (rational)

7 0
2 years ago
What is the length of a side cube with a volume of 729m3
shusha [124]
27 because said a cube teacher told me to do s time s that means square•square
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3 years ago
Shelby made 5quarts of juice. she said that she made 1 1\4of juice.expkain her mistake
docker41 [41]
Shelby mistake is 11;4 is not the correct mixed number
8 0
3 years ago
45 1/3 x 8 1/9 = m<br> m = ?<br> If you answer I will rate you 5 stars. Thank you!
Nastasia [14]

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Step-by-step explanation:

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7 0
2 years ago
The amount of money that Li spends on a video game depends on how many game he purchases. if Li has at most $90 to spend on game
siniylev [52]

Answer:

Hence Domain set for function is {0,1,2,3,4}

Step-by-step explanation:

Given:

Total amount spend =90

22.5 $ each game cost

To Find:

What is domain for the function

Solution:

Now domain mean all possible value for input so that it gives real output as per condition mentioned

So Our condition is that li spends money on video game depends on how many game he purchases

so  total money  spent on game is output

i.e in order to get that much amount of games.

If he spends 22.5 $ for each game and Total amount spent is 90

So 90/22.5=4 Now amount of game he would purchase will be 4

i.e range for lil is 0 to 4 wiht 90 $ in hand

So Domain set will be ranging from{0,1,2,3,4}

7 0
3 years ago
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