8. The figure shown below gives the graph of f, answer the following questions f

Question

8. The figure shown below gives the graph of f, answer the following questions f'(x) -6 A a) f has how many inflection points? Pick the correct choice b) f has how many inflection points? Pick the correct choice i)0 ii) I(iii) 2 tv) 3 v)4 )At- f is concave (up or down) d) List the inflection points of f from the labeled points A, B, C, D and E. e) Which of the following statements is NOT true? i) A is a critical point of f i) A s an inflection point of f iA is a critical point of f iv)A is an inflection point of f f find the absolute max and absolute min of f over -6sxS7 at X=2 Abs Max offover [-6, 7] Abs Min of f over [-6, 7]

Explanation / Answer

Inflection points are where the function changes concavity. Since concave up corresponds to a positive second derivative and concave down corresponds to a negative second derivative, then when the function changes from concave up to concave down (or vise versa) the second derivative must equal zero at thatpoint.

(a)

Inflection points of f are where f'' are 0

it means in graph of f'(x) the slope is 0.

which is the case at point A, B, D

so answer is option (iv) 3

(b)

f'(x) inflection point can be directly observed by change in the curvature points

so answer is 2 as there are 3 different curvatures in the graph

so answer is option (iii) 2

(c)

concave up: where function is maxima, second derivative is positive

concave down: where function is minima, second derivative is negative

at x = -1, f'(x) >0 and it is increasing so f''(x) > 0 so it is concave up

(d)

inflection points of f = A, B, D as expplained in the first part itself

(e)

critical point: where first derivative is 0, so A , C and E are critical points.

(i) true, (ii) true (iii) true (because slope of tangent at f'(x) at point A is also 0, so it is critical point of f'(x) as well)

(iv) false (inflection point is somewhere between A and B)

(f)

from x = -6 to x = 7

Abs Max : is at B point (x = 0, y = 5)

Abs Min : is at D point (x = 4, y = -2)

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