Draw the sign diagram for f ' and find the relative extrema of f
Calculus

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1) Let's notice that if we choose any x value less than 3 (e.g. x = 2) and we make a tangent line at x = 2 then we will have a line that is going up (its slope is positive) then this means f '(x<3) = Positive (+).
2) Then if we make a tangent line at x = 3 then we will have a horizontal line (its slope is zero) which means f '(3) = 0. When this happens then it means that it is a relative maximum or minimum. It is a minimum if it concaves up (like a parabola opens up > this way U) and it is a maximum if it concaves down (like a parabola opens down). So at x = 3 we have a relative maximum.
3) Between x = 3 and x = 5 (e.g. at x = 4) we can see that if we make a tangent line at x = 4 then the line goes down (its slope is negative) which means f'(3< x < 5) = Negative ().
4) At x = 5 the tangent line is horizontal, so f'(5) = 0 and since it is concaves up like a U then we have a relative minimum at x = 5.
5) Between x = 5 and x = 7 we can see that the tangent line is going up. So f'(5< x <7) = Positive (+).
6) At x = 7 we can see a change in its concavity, it concaves down first and then it concaves up and its tangent line is horizontal which means f'(7) = 0 (its slope is 0). So when this happens then it means that it is an inflextion point (it is not a maximum and neither a minimum, it's just an inflexion point). So at x = 7 is an inflexion point and f'(7) = 0.
7) Between x = 7 and x = 9 the tangent line is going up (its slope is positive). So f'(7 < x < 9) = Positive (+).
8) At x = 9 the tangent line is horizontal. So f'(9) = 0 and it is a relative maximun at x = 9.
9) f'( 9< x < 12) = Negative (), the tanagent line is going down.
Please let me know if you have a doubt or question.
hi i just need some clarification i have attached the only acceptable answers can you please clarify which one should be for which Screen Shot 20150715 at 1.15.29 AM.png
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