i have the graph of the derivative of f and....?

Question

i have the graph of the derivative of f

the graph seems to make W shape with another loop

it seems to have 4 roots that are on the positive side of the x axis

all of these roots are in order starting with the first nearest to the origin...

the first root is just the line going down from above the x axis to below the x axis.

the second root is the line looping back up from below the x axis but it only touches the x axis and loops back down.... however, in this case, the touch to the x axis is not smooth but is more like a "kink", sharp pointy...

the third root is just the line looping back up again and this time going from the bottom of the x axis, through the x axis to the top of the x axis...

the fourth root is just the line looping back down from the tope of the x axis but smoothly looping back up just after barely touching the x axis... so this time there was no kink or sharpness... just plain smooth loop...

see additional details below...

i would like to know the local minimum and maximum of f (Not the derivative of f, the graph that i have described above)

from my reasoning...

since f'=0 at the extrema, i concluded that the answer would have to be one of the four roots

but i am not sure which ones...

i would say that since the first root is changing from + to -, it must be local maximum

and since the third root is changing from - to + it would have to be the local minimum...

i am not sure about the others...
i am not completely sure about my reasoning also...


thank you soooooo much

i wanna add that i also think that since the last root is going from + to 0 to + again, it might just be an inflection point and thus not a local max or min....

i wanna add that i also think that since the last root is going from + to 0 to + again, it might just be an inflection point and thus not a local max or min....

Answer 1

A few hints for you:

If f'(x1) = is positive, then f(x) is increasing at x = x1 and if f'(x1) is negative, then f(x) is decreasing at x = x1.

The sharp point you mention is due to a discontinuity, and so f(x) is not a smooth function. When f(x) and f'(x) are both continuous you have a smooth function.

If f(x) is a smooth function, and if as x increases through a critical value x =a, the derivative f'(x) changes sign from positive to negative, then f(a) is a maximum value of f(x0; but if f'(x) changes sign from negative to positive, then f(a) is a minimum value of the function.

Maximum: f'(a) = 0 f'(x) changes from + to -
Minimum: f'(a) =0 f'(x) changes from - to +

Answer 2

Well f' is undefined at the "pointy" loop. The local max and min would also be at the rounded parts of the graph because that is where f'=0.



P.S. I MIGHT be wrong