[Horizontal Tangent Lines] Finding the horizontal tangent lines of trig functions

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bbl

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The given problem is:

number 8.png
I know that I have to first find the derivative and then equate that to 0 since the slope of a horizontal tangent line is 0. However, how will I go about looking for the roots if there are trig functions involved? I would like to know how to solve this for future reference as well. Thank you in advance!
 
bbl said:
The given problem is:

View attachment 31462
I know that I have to first find the derivative and then equate that to 0 since the slope of a horizontal tangent line is 0. However, how will I go about looking for the roots if there are trig functions involved? I would like to know how to solve this for future reference as well. Thank you in advance!
Click to expand...
What would be an equation of a horizontal line? What would be the slope of that tangent line?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:


Please share your work/thoughts about this problem.
 
Subhotosh Khan said:
What would be an equation of a horizontal line? What would be the slope of that tangent line?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:


Please share your work/thoughts about this problem.
Click to expand...
My bad, I'll edit it now.
 
bbl said:
The given problem is:

View attachment 31462
I know that I have to first find the derivative and then equate that to 0 since the slope of a horizontal tangent line is 0. However, how will I go about looking for the roots if there are trig functions involved? I would like to know how to solve this for future reference as well. Thank you in advance!
Click to expand...
I have found the derivative which is -cos(x) + (x)/[(x^2 + 1)^1/2] + 4x^3. However, I got stumped when I equated it to 0 because of the -cos(x). So how do I go about looking for the roots if -cos(x) + (x)/[(x^2 + 1)^1/2] + 4x^3 = 0.
 
bbl said:
Yes, I have found the derivative which is -cos(x) + (x)/[(x^2 + 1)^1/2] + 4x^3. However, I got stumped when I equated it to 0 because of the -cos(x).
Click to expand...
I don't believe you can find the root of that function analytically. You'll need to use numerical approximation methods.
 
Are you expected to use numerical methods here ? If yes - then - what are the methods that you have been taught?

For Newton's method - initial guess could be x = 0.5
 
bbl said:
The given problem is:

View attachment 31462
I know that I have to first find the derivative and then equate that to 0 since the slope of a horizontal tangent line is 0. However, how will I go about looking for the roots if there are trig functions involved? I would like to know how to solve this for future reference as well. Thank you in advance!
Click to expand...
The problem doesn't ask you to find the location of a horizontal tangent, just to prove its existence:

1646277617360.png1646277674469.png

Do you know any theorems that involve existence? What have you been learning recently, if this is for a class?
 
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