Answer:
The true statements are:
4) As x increases on the interval [0, ∞), the rate of change of f eventually exceeds the rate of change of both g and h.
5) A quantity increasing exponentially eventually exceeds a quantity growing quadratically or linearly.
Step-by-step explanation:
f(x)=3^x+2
g(x)=20x+4
h(x)=2x^2+5x+2
1) Over the interval [2, 3], the average rate of change of g is lower than that of both f and h.
Over the interval [a,b], the average rate of change of a function "j" is:
rj=[j(b)-j(a)]/(b-a); with a=2 and b=3
rj=[j(3)-j(2)]/(3-2)
rj=[j(3)-j(2)]/(1)
rj=j(3)-j(2)
For g(x):
rg=g(3)-g(2)
g(3)=20(3)+4→g(3)=60+4→g(3)=64
g(2)=20(2)+4→g(2)=40+4→g(2)=44
rg=64-44→rg=20
For f(x):
rf=f(3)-f(2)
f(3)=3^3+2→f(3)=27+2→f(3)=29
f(2)=3^2+2→f(2)=9+2→f(2)=11
rf=29-11→rf=18
For h(x):
rh=h(3)-h(2)
h(3)=2(3)^2+5(3)+2→h(3)=2(9)+15+2→h(3)=18+15+2→h(3)=35
h(2)=2(2)^2+5(2)+2→h(2)=2(4)+10+2→h(2)=8+10+2→h(2)=20
rh=35-20→rh=15
Over the interval [2, 3], the average rate of change of g (20) is greater than that of both f (18) and h (15), then the first statement is false.
2) As x increases on the interval [0, ∞), the rate of change of g eventually exceeds the rate of change of both f and h.
False, because of f(x) is an exponential function, the rate of f eventually exceeds the rate of change of both g and h.
3) When x=4, the value of f(x) exceeds the values of both g(x) and h(x).
x=4→f(4)=3^4+2=81+2→f(4)=83
x=4→g(4)=20(4)+4=80+4→g(4)=84
x=4→h(4)=2(4)^2+5(4)+2=2(16)+20+2=32+20+2→h(4)=54
When x=4, the value of f(x) (83) exceeds only the value of h(x) (54), then the third statement is false.
4) As x increases on the interval [0, ∞), the rate of change of f eventually exceeds the rate of change of both g and h.
True, because of f(x) is an exponential function.
5) A quantity increasing exponentially eventually exceeds a quantity growing quadratically or linearly.
True.
6) When x=8, the value of h(x) exceeds the values of both f(x) and g(x).
x=8→f(8)=3^8+2=6,561+2→f(8)=6,563
x=8→g(8)=20(8)+4=160+4→g(8)=164
x=8→h(8)=2(8)^2+5(8)+2=2(64)+40+2=128+40+2→h(8)=170
When x=8, the value of h(x) (170) exceeds only the value of g(x) (164), then the sixth statement is false.
