Please answer with steps along with sketch of approximated f(x) and f \"(x) of g

Question

Please answer with steps along with sketch of approximated f(x) and f "(x) of graph. Thank you

The graph below is the graph of f', the derivative off The domain of the derivative is -5sxs 6. -5 -3-1 answers s The critical points for f are x = The critical points for f, are x= j has a local maximum when x /has its maxínum value on [-5,6 ] when x = rts is decreasing on the interval(s), The graph of f is concave up on the interval(s) The x-coordinates of the points-of inflection are x- f' has its maximum value when x = f" has its maximum value when x =- Does f' have a minimum value on [-5,6]? Explain. Does f" have a minimum value on [-5,6]? Explain

Explanation / Answer

The critical points for f are the points where f' is 0

so we have total 3 points = (-3.0),(1,0),(3,0)

The critical points for f' are the points where f'' is 0, means where there is a peak eiher maxima or minima

so we have total 3 points 2 maxima and 2 minima at x=-3,-1,2

f has a local maximum when f' changes its sign from positive to negative

so we have total 1 point x=3

when f' is moving from maxima to minima means the value of f is decreasing

so we have x belongs to (-3,-1) and x belongs to (2,4)

f is concave up in the interval means when f' goes from 0 to negative and then back to positive value and then againt to 0

so we have the interval-> x belongs to (-3,3)

f' has a maximum when x=x

f'' has a max when f' goes from minima to maxima and the point after the maxima where f'=0 is the point of maxima for f''

means in our case it is x=3

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