1. Given f(x)=-6/x, choose the correct statement
A. The graph of f is concave upward on the interval (negative infinity, 0)
B. The graph of f is concave downward on the interval (negative infinity, 0)
C. The graph of f is concave upward on the interval (negative infinity, infinity)
D. The graph of f is concave upward on the interval (0, infinity)
E. none of these
I got A. I found the first derivative. I found the second derivative. I used the interval test to determine concavity.
2. Which of the following statements is true of f(x)=-x^3+18x^2-105x+198
A. f is decreasing on the interval (6, infinity)
B. f is decreasing on the interval (5, 7)
C. f is decreasing on the interval (negative infinity, 5)
D. f is increasing on the interval (5, 7)
E. none of these
I got D. I found the first derivative and critical numbers. I used the interval test to determine increasing or decreasing.
3. Find all points of extrema on the interval [0,2pi] if y=x-cosx
A. (-1.-1+(3pi/2))
B. (pi, pi+1)
C. (-1, 0)
D. ((3pi/2), (3pi/2))
E. none of these
I got D. I found the derivative and when sinx=-1. I got that number and plugged it into the original function to find the point.
Thank you for checking my answers.
#1 ok
#2
f' = -3x^2+36x-105
= -3(x^2-12x+35)
= -3(x-5)(x-7)
f'>0 on (5,7), so D is correct
#3
y' = 1+sinx
y'=0 only at x = 3pi/2, so (D) is correct
Your answers are correct!
1. To determine the concavity of the graph of f(x)=-6/x, you correctly found the first and second derivatives. By using the interval test, you determined that the graph is concave upward on the interval (negative infinity, 0). Therefore, the correct statement is A.
2. To determine where the function f(x)=-x^3+18x^2-105x+198 is increasing or decreasing, you found the first derivative and identified the critical numbers. Using the interval test, you correctly determined that f is increasing on the interval (5, 7). Therefore, the correct statement is D.
3. To find the points of extrema on the interval [0,2pi] for the function y=x-cosx, you found the derivative and determined when sinx=-1. By solving sinx=-1, you correctly found the point (3pi/2, 3pi/2), which is the maximum point. Therefore, the correct statement is D.
Well done! Your answers are correct.
You're welcome! I am here to help. Let's go through each question together to verify your answers:
1. Given f(x)=-6/x, and you concluded that the graph of f is concave upward on the interval (negative infinity, 0). To determine the concavity of a function, you correctly used the second derivative test. However, it seems like you made an error in your answer choice. The correct statement should be:
D. The graph of f is concave downward on the interval (0, infinity)
Please recheck your calculations for finding the concavity using the second derivative test.
2. For the function f(x) = -x^3 + 18x^2 - 105x + 198, you determined that the function is increasing on the interval (5, 7). Again, you correctly used the first derivative test to find the increasing or decreasing nature of the function. Therefore, your answer is correct:
D. f is increasing on the interval (5, 7)
Great job on this one!
3. You were asked to find all points of extrema on the interval [0, 2pi] for the function y=x-cosx. You found the derivative correctly and set it equal to zero to find the critical points. Then, by substituting those critical points back into the original equation, you determined the points of extrema. However, it seems like there is an error in your answer choice. The correct statement should be:
B. (pi, pi + 1)
Please recheck your calculations to find the correct extrema on the given interval.
Overall, you did a decent job solving these questions! Just double-check the specified answers to ensure their accuracy.