1. Find all points of inflection of the function f(x)=x^4+x^3
A. (0, 0) and (-½, -1/16)
B. (-½, -1/16)
C. (0, 0)
D. (0, 0) and (-¾, -27/256)
E. none of these
I got A. I found the second derivative and solved for x and plugged in the values to find the points.
2. Given that f(x)=-x^2+18x-78 has a relative maximum at x=9, choose the correct statement.
A. f’ is negative on the interval (negative infinity, 9)
B. f’ is negative on the interval (9, infinity)
C. f’ is positive on the interval (negative infinity, infinity)
D. f’ is positive on the interval (9, infinity)
E. none of these
I got B. I found the first derivative and critical number. I used the interval test and first derivative test to determine positive or negative.
3. Let f”(x)=4x^3-2x and let f(x) have critical values -1, 0, and 1. Determine which critical values give a relative maximum
A. -1
B. 0
C. 1
D. -1 and 1
E. none of these
I got A. I put the critical values into intervals and tested them to determine positive or negative
Thank you for checking my answers..
All 3 are correct
You're welcome! I'm glad you found my explanations helpful. Now let's go through each question and see if your answers are correct:
1. Find all points of inflection of the function f(x) = x^4 + x^3.
To find the points of inflection, we need to check the concavity of the function. We can do this by finding the second derivative of f(x).
First derivative: f'(x) = 4x^3 + 3x^2
Second derivative: f''(x) = 12x^2 + 6x
To find the points of inflection, we need to equate f''(x) to zero and solve for x:
12x^2 + 6x = 0
2x(6x + 3) = 0
Setting each factor to zero, we get:
2x = 0 --> x = 0
6x + 3 = 0 --> 6x = -3 --> x = -1/2
So, the correct answer is D. (0, 0) and (-½, -1/16).
2. Given that f(x) = -x^2 + 18x - 78 has a relative maximum at x = 9, choose the correct statement.
To determine the sign of the derivative around the critical point, we need to find the first derivative of f(x).
First derivative: f'(x) = -2x + 18
If f(x) has a relative maximum at x = 9, then the slope of the tangent line at that point should be negative. Therefore, f'(9) should be negative.
Let's check the options:
A. f’ is negative on the interval (negative infinity, 9)
B. f’ is negative on the interval (9, infinity)
C. f’ is positive on the interval (negative infinity, infinity)
D. f’ is positive on the interval (9, infinity)
Since we are looking for a negative value at x = 9, the correct answer is B. f’ is negative on the interval (9, infinity).
3. Let f''(x) = 4x^3 - 2x, and let f(x) have critical values -1, 0, and 1. Determine which critical values give a relative maximum.
To determine if the critical points correspond to a relative maximum or minimum, we can analyze the sign of the second derivative in their respective intervals.
The critical points are -1, 0, and 1. Let's plug these values into the second derivative.
When x = -1:
f''(-1) = 4(-1)^3 - 2(-1) = 4 + 2 = 6, which is positive.
When x = 0:
f''(0) = 4(0)^3 - 2(0) = 0, which is neither positive nor negative.
When x = 1:
f''(1) = 4(1)^3 - 2(1) = 4 - 2 = 2, which is positive.
Since the second derivative is positive for both -1 and 1, these critical values give a relative maximum.
Therefore, the correct answer is A. -1.
Based on your explanations, all your answers are correct! Well done! If you have any more questions, feel free to ask.