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
lara31 [8.8K]
3 years ago
13

Consider the following. f(x) = x5 − x3 + 6, −1 ≤ x ≤ 1 (a) Use a graph to find the absolute maximum and minimum values of the fu

nction to two decimal places. maximum minimum (b) Use calculus to find the exact maximum and minimum values. maximum minimum

Mathematics
1 answer:
kykrilka [37]3 years ago
4 0

ANSWER

See below

EXPLANATION

Part a)

The given function is

f(x) =  {x}^{5}  -  {x}^{3}  + 6

From the graph, we can observe that, the absolute maximum occurs at (-0.7746,6.1859) and the absolute minimum occurs at (0.7746,5.8141).

b) Using calculus, we find the first derivative of the given function.

f'(x) = 5 {x}^{4} - 3 {x}^{2}

At turning point f'(x)=0.

5 {x}^{4} - 3 {x}^{2}  = 0

This implies that,

{x}^{2} (5 {x}^{2}  - 3) = 0

{x}^{2}  = 0 \: or \: 5 {x}^{2}  - 3 = 0

x =  - \frac{ \sqrt{15} }{5}   \: or \: x = 0 \:  \: or \: x =\frac{ \sqrt{15} }{5}

We plug this values into the original function to obtain the y-values of the turning points

(   -  \frac{ \sqrt{15} }{5}  , \frac{1}{125} ( 6 \sqrt{15}  +750)) \:and \:  (0, - 6) \: and\: (   \frac{ \sqrt{15} }{5}  , \frac{1}{125} ( - 6 \sqrt{15}  +750))

We now use the second derivative test to determine the absolute maximum minimum on the interval [-1,1]

f''(x) = 20 {x}^{3}  - 6x

f''( -  \frac{ \sqrt{15} }{5} ) \:   <  \: 0

Hence

(   -  \frac{ \sqrt{15} }{5}  , \frac{1}{125} ( 6 \sqrt{15}  + 750))

is a maximum point.

f''( \frac{ \sqrt{15} }{5} ) \:    >  \: 0

Hence

(     \frac{ \sqrt{15} }{5}  , \frac{1}{125} (- 6 \sqrt{15}  + 750))

is a minimum point.

f''(0) \: =\: 0

Hence (0,-6) is a point of inflexion

You might be interested in
What is the period of the graph of y = 5cos(6x – 1) + 3?<br> O A.
evablogger [386]

Step-by-step explanation:

The coefficent of the x variable is b.

The formula for the period is

\frac{2\pi}{ |b| }

Here, b is 6 so we have

\frac{2\pi}{ |6| }  =  \frac{\pi}{3}

So the period is pi/3, or 60 degrees

5 0
2 years ago
Easy Geometry Riddle
Stella [2.4K]

Answer:

A postage stamp

Step-by-step explanation:

A postage stamp sits in the corner of an envelope, yet the envelope that it's on can travel around the world.

~theLocoCoco

3 0
3 years ago
Read 2 more answers
Can someone please help me with numbers 1, a, b, c, 2, a, b, c
algol [13]
Thank you thank you very much
7 0
3 years ago
Name The postulate or theorem you can use to prove the triangles congruent
Zielflug [23.3K]
The answer is HL. HL is a theorem stating that if the hypotenuse and one leg are congruent to another pair of hypotenuse and leg then the triangles themselves are congruent. Hope this helps. 
8 0
3 years ago
What is the surface area of a sphere with the radius of 3?
inessss [21]

Answer:

A≈113.1

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
Other questions:
  • Every used to make $22 an hour, but got a 10% raise. How much more will he make in a 40 hour work week with the raise?
    15·1 answer
  • Use mental math to approximate what a 15% tip would be for a bill of $31.99.
    9·2 answers
  • A bag contains 8 purple beads 2 blue beads and 2 pink beads. The probability of randomly drawing a pink beads is 1/6 . What is t
    15·1 answer
  • Eli practices for a piano recital 3 3 /4 hours in one week. In the same week, he practices basketball 1 2/ 3 hours. How much lon
    12·2 answers
  • If you were to place $2,500 in a savings account that pays 3% interest
    5·1 answer
  • No stealing points!!!!!
    14·1 answer
  • This circle is centered at the origin, and the length of its-radius is 3. What is
    7·1 answer
  • Is this a function yes or no? right answers only please!
    9·2 answers
  • Mr. gonzalez earns his living as a salary plus commission employee. his annual salary is $18,000. he makes 4% commission on all
    10·1 answer
  • Which of the following illustrates the associative property of multiplication?
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!