Find the inflection point(s), if any, of the function. (If an answer does not exist, enter DNE.)
g(x)=9x^3-4x
as you know, inflection points occur where g'' = 0
g'' = 54x
so, at x=0.
so is the point (0,0)?
To find the inflection point(s) of the function g(x) = 9x^3 - 4x, we need to determine the second derivative of the function and find the x-values where it equals zero.
1. Find the first derivative of g(x):
g'(x) = d/dx (9x^3 - 4x)
= 27x^2 - 4
2. Find the second derivative of g(x) by differentiating g'(x):
g''(x) = d/dx (27x^2 - 4)
= 54x
3. Set the second derivative equal to zero and solve for x:
54x = 0
x = 0
The second derivative is equal to zero at x = 0, which means there may be an inflection point at x = 0.
To determine if there is a change in concavity at this point, we can examine the concavity of the function on either side of x = 0.
4. Pick a value less than 0, say x = -1, and substitute it into the second derivative:
g''(-1) = 54(-1) = -54
The second derivative is negative for x = -1, indicating that the function is concave down.
5. Pick a value greater than 0, say x = 1, and substitute it into the second derivative:
g''(1) = 54(1) = 54
The second derivative is positive for x = 1, indicating that the function is concave up.
Since the concavity changes from concave down to concave up at x = 0, we can conclude that there is an inflection point at x = 0.
Therefore, the function g(x) = 9x^3 - 4x has one inflection point at x = 0.
To find the inflection point(s) of the function g(x) = 9x^3 - 4x, we need to determine the points where the concavity of the function changes.
To do this, we will take the second derivative of g(x) and set it equal to zero, and then solve for x. If the second derivative is undefined at a certain x-value, it means that there is no inflection point at that specific x-value, so we also need to check for any points where the second derivative may be undefined.
First, let's find the first derivative of g(x):
g'(x) = d/dx (9x^3 - 4x)
= 27x^2 - 4
Now, let's find the second derivative of g(x):
g''(x) = d/dx (27x^2 - 4)
= 54x
Setting g''(x) equal to zero to find possible inflection points:
54x = 0
x = 0
Taking the second derivative shouldn't give us any undefined points, as there are no restrictions or specific conditions mentioned in the problem.
Therefore, the only possible inflection point of the function g(x) = 9x^3 - 4x is at x = 0.
So, the inflection point of the function is (0, g(0)). To find the y-coordinate of the inflection point, we can substitute x = 0 into the original function:
g(0) = 9(0)^3 - 4(0)
= 0 - 0
= 0
Hence, the inflection point of the function g(x) = 9x^3 - 4x is (0, 0).