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
The product of 9 and number (x) is 88. Which equation shows this relationship?
Digiron [165]
This equation is
9*x =88
5 0
3 years ago
A reasonable estimate of the
lyudmila [28]

Answer:

t

f

t

t

f

f

Step-by-step explanation:

6 0
2 years ago
There are 13 apples in a basket 6 of these apples are green the rest of them are red
kolbaska11 [484]
7 apples are red because 13-6=7

3 0
3 years ago
Aaron has $4,500 saved to pay for an ATV that costs $7,500.
Komok [63]

Answer:

i think its 0.9994225

Step-by-step explanation:

just subtract the 8.75136.24 bye 7.75193. and there you go.

6 0
3 years ago
you are asked to find the mean of the data shown in the stem and leaf plot. which tools are appropriate?
Elina [12.6K]
<span>paper and pencil to record work</span>
8 0
3 years ago
Read 2 more answers
Other questions:
  • A wave has a frequency of 100 Hz and a wavelength of 0.30 m. How do you find the wave speed?
    15·1 answer
  • Find the product. Simplify, if needed. 2 1/8×1 1/6×3
    11·1 answer
  • What us tge distributive property if 14+63
    11·1 answer
  • If the outliners are not included in the data set below, what is the mean of the data set?
    14·1 answer
  • Triangle ABC is rotated 180° using the origin as the center of rotation.
    13·1 answer
  • Describe how to regroup to solve the subtraction problem below what is the difference 456-238
    10·1 answer
  • Write the model of a 50 gram substance that decays at a rate of 3% yearly
    6·1 answer
  • What’s the mean,median,mode and range
    6·1 answer
  • Given the equation below, if x = 4, what is the value of y? y=14x
    6·1 answer
  • (x+3)(3x² - 5x - 10) multiply polynomial
    11·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!