1. Which of the following describes the behavior of f(x)=x^3-x
A. Relative maximum: x=0
B. Relative maximum: x=(1/sqrt(3)); Relative minimum: x=(-1/sqrt(3))
C. Relative maximum: x=(-1/sqrt(3)); Relative minimum: x=(1/sqrt(3))
D. Relative minimum: x=0
E. none of these
I got C. I found the first derivative and critical numbers. I used the interval test and first derivative test to determine max or min.
2. f(x)=(x+2)^3-4. The point (-2, -4) is which of the following?
A. An absolute maximum
B. An absolute minimum
C. A critical point but not an extremum
D. Not a critical point
E. None of these
I got B. I found the first derivative and critical numbers. I plugged in my critical number into the original function and found it matched the given point meaning it is a max or min. I used the interval test and first derivative test to determine max or min.
3. State why Rolle’s Theorem does not apply to f(x)=x^⅔ on the interval [-1, 1]
A. f is not continuous on [-1, 1]
B. f is not defined on the entire interval
C. f is not differentiable at x=0
D. f(1) does not equal f(-1)
E. none of these
I got B. I plugged in -1 and 1 into the function to see if they equalled and they did. The domain I got was [0, infinity). That means the function isn't defined on the entire interval.
Thank you for checking my answers.
correct.
Well done, student!
These are model posts of homework problems. The problem is clearly written, possible answers are given (optional, of course), and the student's answer is noted, along with an indication of the solution method. If actual work had been shown, that would have been nice instead.
Also, only a few problems were posted at once. We tutors get so tired of a marathon post with 20 questions, or twenty successive postings by the same person (or aliases), with no indication of effort by the student. Who likes a moocher? Or homework dumper?
Finally, it has been evident that some effort has been expended, as later postings included the explanation of solution method, and more of the problems have been correctly solved. Long sets of problems or gobs of postings with no work make it very tedious for the tutors. The ideal situation is for a problem or two to be posted, solutions or hints provided, and then the student only needs to come back with more problems if there's still difficulty.
The idea, after all, is to learn to solve the problems. All the freebies in the world here will not help much at test time, when you're on your own.
Your answers are mostly correct.
1. The correct answer is B. To find the behavior of the function f(x)=x^3-x, you correctly found the first derivative, which is f'(x) = 3x^2 - 1. Then, you found the critical numbers by setting the derivative equal to zero: 3x^2 - 1 = 0. Solving this equation, you get x = ±√(1/3). From the first derivative test, you determined that x = (√(1/3)) is a relative minimum, and x = - (√(1/3)) is a relative maximum. So the correct answer is B.
2. The correct answer is B. You correctly found the first derivative of f(x) = (x+2)^3 - 4, which is f'(x) = 3(x+2)^2. The critical number is x = -2. To determine if it is a maximum or minimum, you checked the value of f(-2) = (-2+2)^3 - 4 = -4, which matches the given point (-2, -4). Therefore, (-2, -4) is an absolute minimum. So the correct answer is B.
3. The correct answer is C. Rolle's Theorem states that for a function to satisfy the conditions of the theorem, it must be continuous on the closed interval [-1, 1] and differentiable on the open interval (-1, 1). In this case, the function f(x) = x^⅔ is not differentiable at x = 0 because the exponent is less than 1, which violates the differentiability requirement. Therefore, Rolle's Theorem does not apply. So the correct answer is C.
Well done on your explanations and reasoning!