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Question

2.) For the function f(x), you are given that f'(x) exists for all x and:

i.) f'(0)=0,f"(0)<0,f(0)=10

ii.)f'(-3)=0, f"(-3)>0,f(-3)=-9

iii.) f"(-2)=0

f"(-2)<0 for x>-2

f"(-2)>0 for x<-2

f"(-2)=0

iv.) f(-5)=f(-3)=f(1)=0

A.) Identify the critical points.

B.)Determine the intervals where f(x) is increasing.

C.) Identify any points of inflection.

D.) Determine the intervals where f(x) inconcave up and concave down.

E.) Sketch graph

Answer 1

A, C and D) Critical points. (0,10) is a maximum (i), which will allow the curve to change direction to intersect the x axis at x=1 (iv); (-3,-9) is a minimum (ii), although (iv) gives x=-3 as a zero; point of inflexion at x=-2 and this goes from concave down to concave up as x moves from less than -2 to greater than -2 (iii); the zeroes are x=-5, -3, 1 (iv); but note again that (ii) and (iv) disagree on f(-3), having f(-3)=both -3 and 0. This may be a misprint.

B) f(x) appears to increase when x>-2 to intersect the f(x) axis at f(x)=10, the local maximum. At x=-2, the curve levels off at the point of inflexion. f(x) also appears to continue to increase for x<-5 and then again between two zeroes at x=-5 and -3. After the point of inflexion at x=-2 the curve moves upward through the maximum at (0,10) to cut the x axis again at x=1 and continue with f(x)<0.

C) See above (A, C and D).

D) See above (A, C and D).

E) See B.