▶️ Answer/Explanation
Step 1: Analyze critical points
Given f has exactly two critical points in (10,14), we can deduce:
- The function must change direction twice (from decreasing to increasing or vice versa)
- From the table, we see:
- f decreases from x=10 to x=11 (5→2)
- f increases from x=11 to x=13 (2→6)
- f decreases from x=13 to x=14 (6→5)
Step 2: Evaluate options
A) \( f(x) > 0 \): Not necessarily true – we have no information about values between table points
B) \( f'(x) \) exists: Not necessarily true – critical points could be where derivative doesn’t exist
C) \( f'(x) < 0 \) on (10,11): While true for these discrete points, we can’t guarantee for all x in the interval
D) \( f'(12) \neq 0 \): Must be true because:
- If f'(12)=0, this would be a third critical point (between the min at x≈11 and max at x≈13)
- But we’re told there are exactly two critical points
Conclusion:
Only option D must be true given the conditions.
✅ Answer: D) \( f'(12) \neq 0 \)
Question
\(f(x)=1+x^{\frac{2}{3}}\) , which of the following is NOT true?
(A) f is continuous for all real numbers.
(B) f has a minimum at x = 0 .
(C) f is increasing for x > 0 .
(D) f ′( x) exists for all x.
(E) f ′′(x ) is negative for x > 0 .
▶️ Answer/Explanation
Step 1: Analyze the function properties
Given \( f(x) = 1 + x^{2/3} \), we examine each statement:
A) Continuity: True
\( x^{2/3} \) is defined and continuous for all real numbers.
B) Minimum at x=0: True
\( f(0) = 1 \), and \( f(x) \geq 1 \) for all x since \( x^{2/3} \geq 0 \).
C) Increasing for x>0: True
First derivative \( f'(x) = \frac{2}{3}x^{-1/3} > 0 \) for x > 0.
D) Differentiability: False
\( f'(x) = \frac{2}{3}x^{-1/3} \) is undefined at x=0 (vertical tangent).
E) Concavity: True
Second derivative \( f”(x) = -\frac{2}{9}x^{-4/3} < 0 \) for x > 0.
Step 2: Conclusion
Statement D is the one that is NOT true, as the derivative does not exist at x=0.
✅ Answer: D) f'(x) exists for all x
Question
The graph of \(f ′\) , the derivative of f , is shown in the figure above. Which of the following describes all relative extrema of f on the open interval (a b,)?
(A) One relative maximum and two relative minima
(B) Two relative maxima and one relative minimum
(C) Three relative maxima and one relative minimum
(D) One relative maximum and three relative minima
(E) Three relative maxima and two relative minima
▶️ Answer/Explanation
Question
Let g be a continuous function on the closed interval [0,1]. Let (0)= 1 and g (1) =0 . Which of the following is NOT necessarily true?
(A) There exists a number h in [0,1] such that \(g(h)\geq g(x) \)for all x in [0,1].
(B) For all a and b in [0,1], if a =b , then g(a)=g(b).
(C) There exists a number h in [0,1] such that \(g(h)=\frac{1}{2}\)
(D) There exists a number h in [0,1] such that \(g(h)=\frac{3}{2}\)
(E) For all h in the open interval \((0,1),\lim_{x\rightarrow h}g(x)=g(h)\)