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Homework answers / question archive / Quiz 3, Math-140

Quiz 3, Math-140.

1. (40%)True or False

1. _____ If f’(x) =0 when x=c then f has either a minimum or maximum at x=c.

2. _____ If a differentiable function f has a minimum or maximum at x=c, then f’(c)=0.

3. _____ If f is continuous on an open interval (a, b) then the f attains maximum or

minimum in (a, b).

4. _____ If f’(x)=g’(x) then f(x) = g(x).

5. _____ If x=c is an inflection point for f, then f (c) must be a maximum or minimum of

f.

6. _____ f(x) = ax 2 +bx +c, (with a ≠ 0), can have only one critical point.

7. _____ Second Shape Theorem includes the converse of First Shape Theorem.

8. _____ If f(x) has a minimum at x=a, then there exists an ε, such that f(x) > f(a) for

every x in (a- ε, a+ ε).

9. _____ The mean value theorem applies as long as the function is continuous on an

interval [a, b].

10. _____ If f (x) has an extreme value at x=a then f is differentiable at x=a.

11. ______If f(x) is continuous everywhere, and f(a)=f(b), then there exists x=c such that

f’(c)=0.

12. ______ if f(x) is continuous and differentiable everywhere, then f attains a max or

min at x=a, if f’(a)=0.

13. ______The function f (x) =x 3 does not have an extreme value over the closed interval

[a, b].

14. ______If f(x) is not differentiable at x=a, then (a, f(a)) cannot be an extreme value of

f.

15. ______If f”(a-ε)*f”(a+ε) <0, for an arbitray positive number ε, then the function f(x)

has an extreme value at x=a.

16. ______If f’(a-ε)*f’(a+ε) <0, for an arbitray positive number ε, then the function f(x)

has an extreme value at x=a whenever f’(a)=0, or f’(a) is undefined.

17. ______The function y=(x-2) 3 + 1 has an inflection point at (2, 1)

18. ______The function: is always increasing in (0, +).

19. ______If f’(a)=0, and f”(a)>0 then f has a minimum at x=a.

20. ______If f”(x)>0 on an interval I then f(x) is increasing on I.

2. (20%) Find the maximum and minimum values attained by the function:

3. (20%) Giving a vertical velocity v(t)= 4t +3 (ft/Sec) for a helicopter t seconds after taking

off. The helicopter was observed 17 feet above the sea level, after taking off for 2

seconds. Find a formula that gives the height of the helicopter at any time t? (Hint:

v(t)=h’(t) where h(t) is the height of the helicopter at any time t.)

4. (20%) Consider yourself an entrepreneur at the Mall selling 3000 donuts a day at $0.75 each.

When you raised the price to $1.00 each, the sale dropped to 2500 donuts per day. Now,

assume that there is a linear relationship between the price and number of donuts sold. Further

assume that there is a fixed cost (overhead) of $200 per day, and the cost of each donut is 25¢.

What would be the price of the donut to maximize the profit? (Hint: Since we assume a linear

relationship between price, p, and the number of dounats sold, n, we need first to determine

the function n(p), n(p) = a +b*p. Then, determine an optimal unit price that maximizes the

profit.)