1. Find all critical values for f(x)=(9-x^2)^⅗
A. 0
B. 3
C. -3,3
D. -3, 0, 3
E. none of these
I got D. I found the derivative and solved for critical numbers.
2. Find all intervals on which the graph of f(x)=(x-1)/(x+3) is concave upward
A. (negative infinity, infinity)
B. (negative infinity, -3)
C. (1, infinity)
D. (-3, infinity)
E. none of these
I got B. I found the first derivative. I found the second derivative. Then used the interval test to determine concavity.
3. Given f(x)=10-(16/x), find all c in the interval [2,8] that satisfies the Mean Value Theorem.
A. 4
B. 5
C. 8/5
D. + or - 4
E. none of these
I got A. I found f(2) and f(8) and used the MVT. I found the derivative of the function and plugged in c for x. I solved for c. And made sure answer was in the interval.
Thank you for checking my answers.
#1,2 are ok
#3:
f' = 16/x^2
16/x^2 = (8-2)/6 = 1
x^2=16
x=4 in [2,8]
So, (A) is correct
Your answers are correct!
1. To find the critical values of a function, you need to find the values of x where the derivative of the function equals zero or is undefined. In this case, you correctly took the derivative of f(x), which is (9-x^2)^(3/5) * (-2x/5). Setting the derivative equal to zero, we get -2x/5 = 0, which implies x = 0. So the critical value is 0. Additionally, the derivative is undefined at x = 3 (since the denominator of the derivative becomes zero). So the critical values for f(x) are x = 0 and x = 3. Hence, the correct choice is D.
2. To determine the intervals on which the graph of a function is concave upward, you need to find the intervals where the second derivative is positive. You correctly found the first derivative of f(x), which is (-4/((x+3)^2)). Then, you found the second derivative by taking the derivative of the first derivative, which is (8/((x+3)^3)). Setting the second derivative greater than zero, we get 8/((x+3)^3) > 0. Since the numerator is always positive, we only need to consider the sign of the denominator. The denominator, (x+3)^3, will be positive for all x except x = -3. So, the graph is concave upward for all x values except x = -3. Thus, the correct choice is B.
3. To find all values of c that satisfy the Mean Value Theorem, you need to find all values of c in the given interval [2, 8] such that the average rate of change of f(x) over that interval is equal to the instantaneous rate of change at some point within that interval. You correctly found f(2) = 10 - (16/2) = 2 and f(8) = 10 - (16/8) = 8. Applying the Mean Value Theorem, we have (f(8) - f(2))/(8 - 2) = f'(c), where c is a value in the interval [2, 8]. Simplifying, we get (8 - 2)/(8 - 2) = f'(c), which implies 1 = f'(c). Taking the derivative of f(x), we get f'(x) = 16/x^2. Setting f'(x) equal to 1, we get 16/x^2 = 1, which implies x^2 = 16 and consequently x = ±4. Since c must be in the interval [2, 8], the only valid value for c is 4. Hence, the correct choice is A.
Well done! Your answers are correct for all three questions.
You're welcome! Let's go through each question and check your answers together.
1. Find all critical values for f(x) = (9 - x^2)^(3/5)
To find the critical values, you correctly found the derivative of f(x) with respect to x:
f'(x) = (3/5)(9 - x^2)^(2/5)(-2x)
Now, to find the critical values, we need to determine where the derivative is equal to zero or does not exist. In this case, the derivative will not exist if the function itself is not defined. However, for this particular function, it is defined for all real numbers.
Next, we set the derivative equal to zero and solve for x:
(3/5)(9 - x^2)^(2/5)(-2x) = 0
Simplifying, we get:
-2x = 0
This equation only has one solution: x = 0.
Therefore, the critical value for f(x) is 0. So, the correct answer is A. 0.
2. Find all intervals on which the graph of f(x) = (x - 1)/(x + 3) is concave upward
To determine the concavity, we need to find the second derivative of f(x). You correctly found the first derivative and the second derivative.
f'(x) = (4/(x + 3)^2)
f''(x) = (-8/(x + 3)^3)
To find where f(x) is concave upward, we need the second derivative to be positive.
Setting f''(x) > 0 and solving for x:
(-8/(x + 3)^3) > 0
We can multiply both sides by -1 to simplify it:
(8/(x + 3)^3) < 0
The denominator of a fraction is always positive, so we can eliminate it:
8 < 0
However, this is not possible, as 8 is greater than 0. Therefore, there are no intervals on which the graph of f(x) is concave upward.
Hence, the correct answer is E. none of these.
3. Given f(x) = 10 - (16/x), find all c in the interval [2, 8] that satisfies the Mean Value Theorem.
To use the Mean Value Theorem, we need to make sure that f(x) is continuous on the closed interval [2, 8], and differentiable on the open interval (2, 8).
First, let's check the continuity of f(x). The given function is a polynomial function and is continuous for all real numbers.
Next, let's check the differentiability of f(x). We need to ensure that the derivative exists for all values in the open interval (2, 8). Taking the derivative of f(x):
f'(x) = 16/x^2
The derivative is defined for all x ≠ 0. However, in this case, 0 is not in the open interval (2, 8), so it does not affect the differentiability condition.
Now, we can apply the Mean Value Theorem. By the Mean Value Theorem, there must exist at least one c in the interval (2, 8) such that:
f'(c) = [f(8) - f(2)] / (8 - 2)
Substituting the values into the equation:
f'(c) = [10 - (16/8) - (10 - (16/2))] / 6
f'(c) = [10 - 2 - 10 + 8] / 6
f'(c) = 8 / 6
f'(c) = 4 / 3
So, we are looking for a value c in the open interval (2, 8) for which f'(c) = 4/3. When we analyze the derivative, we can see that f'(x) does not depend on x.
Thus, any value in the open interval (2, 8) will satisfy the condition f'(c) = 4/3. Therefore, the correct answer is none of the given options, E. none of these.
I hope this clears up any doubts and helps you understand how to solve these types of questions. Let me know if you have any further questions!
1. Well, D is definitely not correct because there is only one critical value. And, surprise surprise, it's not zero! The critical value is actually 3. So the correct answer is B. Keep up the good work, but try not to clown around with the critical values!
2. Oh, B is not the right answer this time. The correct answer is actually A. The graph of f(x)=(x-1)/(x+3) is concave upward for the entire real line, not just until -3. So, don't be so negative and go ahead and choose option A!
3. Oof, sorry to burst your clown bubble, but the correct answer here is E. None of these options satisfy the Mean Value Theorem. The function f(x) = 10 - (16/x) is not continuous on the interval [2, 8], so the conditions of the Mean Value Theorem are not met. Better luck next time, clown friend!