Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To find the local maximum and minimum of the function [tex]\( f(x)=7x+3x^{-1} \)[/tex], we will go through the following steps:
### Step 1: Find the Derivative
First, calculate the first derivative of the function to find the critical points.
The function given is:
[tex]\[ f(x) = 7x + 3x^{-1} \][/tex]
Calculate the first derivative:
[tex]\[ f'(x) = \frac{d}{dx}(7x + 3x^{-1}) \][/tex]
[tex]\[ f'(x) = 7 - 3x^{-2} \][/tex]
### Step 2: Set the First Derivative to Zero
We set the first derivative equal to zero to find the critical points:
[tex]\[ 7 - 3x^{-2} = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ 7 = 3x^{-2} \][/tex]
[tex]\[ 7 = \frac{3}{x^2} \][/tex]
[tex]\[ 7x^2 = 3 \][/tex]
[tex]\[ x^2 = \frac{3}{7} \][/tex]
[tex]\[ x = \pm \sqrt{\frac{3}{7}} \][/tex]
[tex]\[ x = \pm \frac{\sqrt{21}}{7} \][/tex]
So, the critical points are [tex]\( x = \frac{\sqrt{21}}{7} \)[/tex] and [tex]\( x = -\frac{\sqrt{21}}{7} \)[/tex].
### Step 3: Second Derivative Test
Next, calculate the second derivative to classify these critical points:
[tex]\[ f''(x) = \frac{d}{dx} (7 - 3x^{-2}) \][/tex]
[tex]\[ f''(x) = 6x^{-3} \][/tex]
[tex]\[ f''(x) = \frac{6}{x^3} \][/tex]
### Step 4: Evaluate the Second Derivative at Critical Points
Evaluate the second derivative at each critical point to determine if it is a local maximum or minimum.
For [tex]\( x = \frac{\sqrt{21}}{7} \)[/tex]:
[tex]\[ f''\left(\frac{\sqrt{21}}{7}\right) = \frac{6}{\left(\frac{\sqrt{21}}{7}\right)^3} = \frac{6 \cdot 7^3}{21^{3/2}} = \frac{6 \cdot 343}{21\sqrt{21}} = \frac{2058}{21\sqrt{21}} > 0 \][/tex]
Since [tex]\( f''\left(\frac{\sqrt{21}}{7}\right) > 0 \)[/tex], [tex]\( x = \frac{\sqrt{21}}{7} \)[/tex] is a local minimum.
For [tex]\( x = -\frac{\sqrt{21}}{7} \)[/tex]:
[tex]\[ f''\left(-\frac{\sqrt{21}}{7}\right) = \frac{6}{\left(-\frac{\sqrt{21}}{7}\right)^3} = -\frac{6 \cdot 343}{21^{3/2}} = -\frac{2058}{21\sqrt{21}} < 0 \][/tex]
Since [tex]\( f''\left(-\frac{\sqrt{21}}{7}\right) < 0 \)[/tex], [tex]\( x = -\frac{\sqrt{21}}{7} \)[/tex] is a local maximum.
### Step 5: Find Function Values at Critical Points
Calculate the function values at these critical points.
For [tex]\( x = \frac{\sqrt{21}}{7} \)[/tex]:
[tex]\[ f\left(\frac{\sqrt{21}}{7}\right) = 7\left(\frac{\sqrt{21}}{7}\right) + 3\left(\frac{7}{\sqrt{21}}\right) \][/tex]
[tex]\[ f\left(\frac{\sqrt{21}}{7}\right) = \sqrt{21} + \frac{21}{\sqrt{21}} = 2\sqrt{21} \][/tex]
For [tex]\( x = -\frac{\sqrt{21}}{7} \)[/tex]:
[tex]\[ f\left(-\frac{\sqrt{21}}{7}\right) = 7\left(-\frac{\sqrt{21}}{7}\right) + 3\left(-\frac{7}{\sqrt{21}}\right) \][/tex]
[tex]\[ f\left(-\frac{\sqrt{21}}{7}\right) = -\sqrt{21} - \frac{21}{\sqrt{21}} = -2\sqrt{21} \][/tex]
### Final Answer:
- The function [tex]\( f(x) \)[/tex] has a local maximum at [tex]\( x = -\frac{\sqrt{21}}{7} \)[/tex] with value [tex]\( -2\sqrt{21} \)[/tex].
- The function [tex]\( f(x) \)[/tex] has a local minimum at [tex]\( x = \frac{\sqrt{21}}{7} \)[/tex] with value [tex]\( 2\sqrt{21} \)[/tex].
### Step 1: Find the Derivative
First, calculate the first derivative of the function to find the critical points.
The function given is:
[tex]\[ f(x) = 7x + 3x^{-1} \][/tex]
Calculate the first derivative:
[tex]\[ f'(x) = \frac{d}{dx}(7x + 3x^{-1}) \][/tex]
[tex]\[ f'(x) = 7 - 3x^{-2} \][/tex]
### Step 2: Set the First Derivative to Zero
We set the first derivative equal to zero to find the critical points:
[tex]\[ 7 - 3x^{-2} = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ 7 = 3x^{-2} \][/tex]
[tex]\[ 7 = \frac{3}{x^2} \][/tex]
[tex]\[ 7x^2 = 3 \][/tex]
[tex]\[ x^2 = \frac{3}{7} \][/tex]
[tex]\[ x = \pm \sqrt{\frac{3}{7}} \][/tex]
[tex]\[ x = \pm \frac{\sqrt{21}}{7} \][/tex]
So, the critical points are [tex]\( x = \frac{\sqrt{21}}{7} \)[/tex] and [tex]\( x = -\frac{\sqrt{21}}{7} \)[/tex].
### Step 3: Second Derivative Test
Next, calculate the second derivative to classify these critical points:
[tex]\[ f''(x) = \frac{d}{dx} (7 - 3x^{-2}) \][/tex]
[tex]\[ f''(x) = 6x^{-3} \][/tex]
[tex]\[ f''(x) = \frac{6}{x^3} \][/tex]
### Step 4: Evaluate the Second Derivative at Critical Points
Evaluate the second derivative at each critical point to determine if it is a local maximum or minimum.
For [tex]\( x = \frac{\sqrt{21}}{7} \)[/tex]:
[tex]\[ f''\left(\frac{\sqrt{21}}{7}\right) = \frac{6}{\left(\frac{\sqrt{21}}{7}\right)^3} = \frac{6 \cdot 7^3}{21^{3/2}} = \frac{6 \cdot 343}{21\sqrt{21}} = \frac{2058}{21\sqrt{21}} > 0 \][/tex]
Since [tex]\( f''\left(\frac{\sqrt{21}}{7}\right) > 0 \)[/tex], [tex]\( x = \frac{\sqrt{21}}{7} \)[/tex] is a local minimum.
For [tex]\( x = -\frac{\sqrt{21}}{7} \)[/tex]:
[tex]\[ f''\left(-\frac{\sqrt{21}}{7}\right) = \frac{6}{\left(-\frac{\sqrt{21}}{7}\right)^3} = -\frac{6 \cdot 343}{21^{3/2}} = -\frac{2058}{21\sqrt{21}} < 0 \][/tex]
Since [tex]\( f''\left(-\frac{\sqrt{21}}{7}\right) < 0 \)[/tex], [tex]\( x = -\frac{\sqrt{21}}{7} \)[/tex] is a local maximum.
### Step 5: Find Function Values at Critical Points
Calculate the function values at these critical points.
For [tex]\( x = \frac{\sqrt{21}}{7} \)[/tex]:
[tex]\[ f\left(\frac{\sqrt{21}}{7}\right) = 7\left(\frac{\sqrt{21}}{7}\right) + 3\left(\frac{7}{\sqrt{21}}\right) \][/tex]
[tex]\[ f\left(\frac{\sqrt{21}}{7}\right) = \sqrt{21} + \frac{21}{\sqrt{21}} = 2\sqrt{21} \][/tex]
For [tex]\( x = -\frac{\sqrt{21}}{7} \)[/tex]:
[tex]\[ f\left(-\frac{\sqrt{21}}{7}\right) = 7\left(-\frac{\sqrt{21}}{7}\right) + 3\left(-\frac{7}{\sqrt{21}}\right) \][/tex]
[tex]\[ f\left(-\frac{\sqrt{21}}{7}\right) = -\sqrt{21} - \frac{21}{\sqrt{21}} = -2\sqrt{21} \][/tex]
### Final Answer:
- The function [tex]\( f(x) \)[/tex] has a local maximum at [tex]\( x = -\frac{\sqrt{21}}{7} \)[/tex] with value [tex]\( -2\sqrt{21} \)[/tex].
- The function [tex]\( f(x) \)[/tex] has a local minimum at [tex]\( x = \frac{\sqrt{21}}{7} \)[/tex] with value [tex]\( 2\sqrt{21} \)[/tex].
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
