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
levacccp [35]
2 years ago
6

Find the maxima and minima of the following function:

Mathematics
2 answers:
Fantom [35]2 years ago
7 0

To find the maxima and minima of the function, we need to calculate the derivative of the function. Note, before the denominator is a perfect square trinomial, so the function can be simplified as

\large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f(x) = \frac{x^2 - x - 2}{(x - 3)^2}} \end{gathered}$}

So the derivative is:

  \large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f'(x) = \frac{(2x - 1)(x - 3)^2 - 2(x - 3)(x^2 - x - 2)}{(x - 3)^4} } \end{gathered}$}

Simplifying the numerator, we get:

                 \large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f'(x) = \frac{(x - 3)(-5x + 7)}{(x - 3)^4} = \frac{-5x + 7}{(x - 3)^3} } \end{gathered}$}

The function will have a maximum or minimum when f'(x) = 0, that is,

                  \large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f'(x) = \frac{-5x + 7}{(x - 3)^3} = 0 } \end{gathered}$}

which is true if -5x + 7 = 0. Then x = 7/5.

To determine whether x = 7/5 is a maximum, we can use the second derivative test or the first derivative test. In this case, it is easier to use the first derivative test to avoid calculating the second derivative. For this, we evaluate f'(x) at a point to the left of x = 7/5 and at a point to the right of it (as long as it is not greater than 3). Since 1 is to the left of 7/5, we evaluate:

                    \large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f(1) = \frac{-5 + 7}{(1 - 3)^3} = \frac{2}{-8} < 0} \end{gathered}$}

Likewise, since 2 is to the right of 7/5, then we evaluate:

                                   \large\displaystyle\text{$\begin{gathered}\sf \displaystyle \bf{\frac{-10 + 7}{(2 - 3)^3} = \frac{-3}{-1} > 0} \end{gathered}$}

Note that to the left of 7/5 the derivative is negative (the function decreases) and to the right of 7/5 the derivative is positive (the function increases).

The value of f(x) at 7/5 is:

                               \large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f\left(\tfrac{7}{5}\right) = \frac{\tfrac{49}{25} - \tfrac{7}{5} - 2}{\tfrac{49}{25} - 6 \cdot \tfrac{7}{5} + 9} = -\frac{9}{16} } \end{gathered}$}

This means that \bf{\left( \frac{7}{5}, -\frac{9}{16} \right)} is a minimum (and the only extreme value of f(x)).

\huge \red{\boxed{\green{\boxed{\boldsymbol{\purple{Pisces04}}}}}}

ira [324]2 years ago
4 0
X2 -x-2 / x( x-5x+9)
You might be interested in
Number of whole numbers between 45 and 69 is _________ .<br> (a) 31 (b) 23 (c) 25 (d) 28
tatiyna

The number of whole number is 23 through subtracting the numbers. hope it helps ☺️

8 0
3 years ago
Read 2 more answers
Which Natural Numbers are not Integers?
Genrish500 [490]

Answer:

Step-by-step explanation:

fraction are not integers

6 0
3 years ago
Which linear inequality is represented by the graph?
kompoz [17]
The answer is D since is the only one that has the correct y intercept
4 0
3 years ago
Listen just answer I neeed help, but I don’t
adelina 88 [10]

Answer:

look below

Step-by-step explanation:

I'll go in order from top to bottom

5\frac{2}{3}

1\frac{5}{9}

6\frac{1}{2}

6\frac{1}{3}

5\frac{1}{2}

5 0
3 years ago
A submarine dives 650 feet every 5 minutes and 5,200 feet every 40 minutes.
Olenka [21]

Answer:

Step-by-step explanation:

8 0
2 years ago
Read 2 more answers
Other questions:
  • How many solutions are there for the system of equations shown on the graph? A coordinate plane is shown with two lines graphed.
    8·2 answers
  • An event lasts for 5 hours and 30 min how long did the event last in seconds
    6·2 answers
  • What’s is the opposite of -5
    15·1 answer
  • What is the value of x given that figure ADCB
    13·2 answers
  • I will mark brainly-ist to who ever helps me
    13·2 answers
  • The west room of the White House has a perimeter of 180 feet. If the width is 50 feet, what is the length of the West room?
    13·1 answer
  • Plz help with the correct answer please
    7·1 answer
  • PLEASE HURRY!!!!<br><br> Find the slope of the line.
    10·2 answers
  • I dont know man help please
    6·1 answer
  • HELP QUICK PLEASE MARK RIGHT ANSWER BRAINLIEST
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!