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
Solve Equation <br> x^2 -4x +8 = 0
mrs_skeptik [129]
X=2 plus or minus 2i
4 0
3 years ago
Write an addition fact that will give the same sum as 8+7
ahrayia [7]
You could do 5+10 or 12+3
4 0
3 years ago
Read 2 more answers
Plz help me this is a final!!!
Marat540 [252]

Answer: (a) 10 (b) 1. 10 2. -10 (c) 1. 10 2. 10

Step-by-step explanation:

6 0
3 years ago
Identify the graph of y=x3.
77julia77 [94]

Answer:

2nd Graph

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
7. Use the properties of operations to simplify this algebraic expression. Rewrite the expression by following the directions in
Svetach [21]

Answer:

-x + -13

Step-by-step explanation:

Rewrite: 5(x – 4) + 3x – 9x + 7

Step 1: 5(x + –4) + 3x + –9x + 7

Step 2: 5x + -20 + 3x + –9x + 7

Step 3: 5x + 3x + –9x + -20 + 7

Step 4: -1x + -13

Step 5: -x + -13

3 0
3 years ago
Other questions:
  • A car can travel 32 miles for each gallon of gasoline. The function ​d(x)equals=32x represents the distance​ d(x), in​ miles, th
    7·1 answer
  • If 5.7 moles of zinc metal react with 8.9 moles of silver nitrate, how many moles of silver metal can be formed, and how many mo
    12·1 answer
  • What does 4q+2/8+8y+4/8+8q-7y equal
    12·1 answer
  • What is the rate of change of the line
    5·2 answers
  • Please help anyone know how to do this.
    12·1 answer
  • The Atlanta Journal Constitution conducted a survey of 900 registered voters to determine which voting method they prefer. The A
    9·1 answer
  • Can u pls help me with this question asap ​
    12·2 answers
  • I know I keep asking for help but I literally just got put back in my right grade pls help
    15·1 answer
  • What is 2 divided by 5
    10·2 answers
  • If 6 = m/8, <br> What does "m" equal?
    9·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!