Thank you for the opportunity to help you with your question! I shall do my best to help you out!
I can't give you exact answers because there is no function listed in the question, however, I will outline steps you would take for any equation to find f'(x).
First, to find f'(x) there are a number of ways depending on the function. You would ideally use chain rule, product rule and/or quoient rule for finding f'(x). You may recall that if you were given: f(x) = 3x2 + 4x - 5 that f'(x) = 6x + 4.
To find the derivative of a polynomial, you differentiate each term of the polynomial - in our case, 3x2 differentiated becomes 6x, 4x diferentiated becomes 4, and -5 differentiated becomes 0. We add up the resultant terms, 6x + 4 + 0 = 6x + 4.
Now onto the second part with points and finding tangent line that is horizontal. After differentiating the function you would look for the roots (i.e. when the line is horizontal). Once you have the two roots, those are your two points (i.e. (x1, 0) and (x2, 0)). You would then plug in the roots into the original f(x) equation to solve for y (you will have two y values if you have two roots). You could then use a table of inequalities to figure out the direction before, between and after each root point (i.e. you would want to know what the graph is doing before x1, between x1 and x2 and after x2).
I hope this helps you out. It's hard to do without solid numbers given, but I hope this gives you some guidence to proceed in your homework.Please let me know if you need any clarification. I'm always happy to answer your questions. Thank you!
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