Consider the function f(x) = 2 x + 3 x ^ { -1 }.
Note that this function has no inflection points, but f''(x) is undefined at x=B where
B=
6/x^3(my answer)
For each of the following intervals, tell whether f(x) is concave up (type in CU) or concave down (type in CD).
(-\infty, B):
CU(my answer)
(B,\infty):
CD(my answer)
hello, my answer in inccorect but i don't know which question is wrong
first derivation = 2 - 3 / x²
second derivation = 6 / x³
Second derivation is undefined when:
6 / x³ →∞
6 / x³ →∞ when x = 0
f''(x) is undefined at x = 0
The graph of a function is concave up on interval where f "(x) > 0
The graph of a function is concave down on interval where y = f "(x) < 0
The graph of a function is concave up when:
f "(x) > 0
6 / x³ > 0
x ∈ ( 0 , ∞ )
The graph of a function is concave down when:
6 / x³ < 0
x ∈ ( - ∞ , 0 )
y = 2 x + 3/x Undefined at x = 0
dy/dx = 2 - 3/x^2 also undefined at x = 0
d^2y/dx^2 = +6x/x^4 = 6/x^3
when x is <0 which means <B
d^2/dy^2 <0 so sheds water, concave down
when x>0
d^2y/dx^2 >0 so holds water, concave up
to check definitions use
y = x^2 which we know holds water
dy/dx = 2x
d^2y/dx^2 = +2
To determine whether the function f(x) = 2x + 3x^(-1) is concave up (CU) or concave down (CD) on the given intervals, we need to find the second derivative of the function.
1. Interval (-∞, B):
To find the concavity on this interval, we need to differentiate f(x) twice with respect to x.
f(x) = 2x + 3x^(-1)
f'(x) = 2 - 3x^(-2)
f''(x) = 6x^(-3)
Since x = B, where B = 6/x^3, is undefined, we cannot evaluate f''(x) at this point. Therefore, B does not exist in the function.
Since there is no inflection point, there is no interval (-∞, B), and we cannot determine the concavity on this interval.
2. Interval (B, ∞):
Similarly, we need to differentiate f(x) twice with respect to x.
f(x) = 2x + 3x^(-1)
f'(x) = 2 - 3x^(-2)
f''(x) = 6x^(-3)
In this case, B is not relevant to determine the concavity on the interval (B, ∞) since we can still evaluate f''(x) at any other value of x greater than B.
Since f''(x) = 6x^(-3), which is positive for x > B, the function f(x) is concave down (CD) on the interval (B, ∞).
In conclusion:
(-∞, B) - Concavity cannot be determined as B is undefined.
(B, ∞) - f(x) is concave down (CD).
Please let me know if you need further clarification or assistance.
To determine whether a function is concave up or concave down, you need to analyze the second derivative of the function.
Let's start by finding the second derivative of the function f(x) = 2x + 3x^(-1):
f(x) = 2x + 3/x
First, find the first derivative using the power rule and quotient rule:
f'(x) = 2 - 3/x^2
Now, find the second derivative by differentiating f'(x):
f''(x) = 6/x^3
Now, we need to find the value of x where f''(x) is undefined. From the given statement, we have B = 6/x^3.
However, there seems to be a misunderstanding in the provided equation to find B. The correct equation to determine where f''(x) is undefined would be:
x^3 = 6
Taking the cube root of both sides, we get:
x = ∛6
So, B is not equal to 6/x^3; rather, B is equal to the cube root of 6 (∛6).
Now, let's determine the concavity of the function on the given intervals:
(-∞, B):
For this interval, since B = ∛6, we can conclude that f''(x) is positive (6/x^3 > 0), which means the function is concave up. Therefore, the answer is CU (Concave Up).
(B, ∞):
For this interval, since B = ∛6, we can conclude that f''(x) is negative (6/x^3 < 0), which means the function is concave down. Therefore, the answer is CD (Concave Down).
In summary, the correct answers are:
(-∞, B): CU (Concave Up)
(B, ∞): CD (Concave Down)
I apologize for the confusion caused by the incorrect equation provided to find B.