Answer the questions regarding the original function f(x). (a) Determine the int

Question

Answer the questions regarding the original function f(x).

(a) Determine the intervals of increase and decrease for the original function f(x).

(b) Find the x-coordinates of all critical points of the original function f(x), and determine their nature (local maximum, local minimum, or neither).

(c) Determine the intervals of concavity for the original function f(x).

(d) Find the x-coordinates of all inection points

Answer the questions regarding the original function f(x). (a) Determine the intervals of increase and decrease for the original function f(x). (b) Find the x-coordinates of all critical points of the original function f(x), and determine their nature (local maximum, local minimum, or neither). (c) Determine the intervals of concavity for the original function f(x). (d) Find the x-coordinates of all inection points

Explanation / Answer

The numbers are not wrrriten on graph, assuming each grid is of 1 unit.

a)

The orignal function f(x) will increase when f'(x) >0.

Since the given graph is of f'(x) not f(x), it will be greater than zero, when graph is above the horizontal axis (or x-axis).

From the given graph, it is above the x-axis in the interval (-4,0)U(0,3)U(5, inf), hence the orignal function will increase in the interval

(-4,0)U(0,3)U(5, inf)

Similarly orignal function will decresae when graph of f'(x) is below the horizontal axis. hence the orignal function will decrease in the interval

(-inf, -4)U(3,5)

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b)

The function has maxima or minima where f'(x) is zero.

Since the given graph is of f'(x), it will be zero when it touches the horizontal axis.

Hence the crtical points are -4,0,3,5.

The function has local minima when graph crosses the x-axis from below, and maxima when crosses the x-axis from above. If it does not cross or touch the x-axis then neither maixma nor minima.

Thus function has minima at x= -4 and x=5

The function has maxima at x=3

The function has neither max or min at x=0

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c)

The function will concave up, when graph of f'(x) is increasing.

Hence the function is concave up in the interval (-inf, -3)U(0,2)U(4,inf)

The function will concave down, when graph of f'(x) is decreasing.

Hence the function is concave down in the interval (-3,0)U(2,4)

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d)

The function has inflection point where slope of f'(x) is zero.(tangent line is horizontal)

Hence the function has inflection point at x= -3,0,2,4

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