3)
f(x)=x^(2)/6x^(2)+4.
List the x values of the inflection points of f.
inflection points where f" = 0
f" = 2(2-9x^2)/(3x+2)^2
f"=0 when 2-9x^2 = 0
To find the inflection points of a function, you need to find the values of x where the concavity of the function changes. The concavity changes at points where the second derivative of the function is equal to zero or does not exist.
Now, let's find the first derivative of f(x) to determine the intervals where the concavity might change:
f(x) = x^2 / (6x^2 + 4)
To find the first derivative, you can use the quotient rule:
f'(x) = (2x(6x^2 + 4) - x^2(12x)) / (6x^2 + 4)^2
Now simplify the expression:
f'(x) = (12x^3 + 8x - 12x^3) / (6x^2 + 4)^2
= (8x) / (6x^2 + 4)^2
The first derivative is f'(x) = 8x / (6x^2 + 4)^2.
Next, find the second derivative by differentiating f'(x):
f''(x) = (8(6x^2 + 4)^2 - 16x(6x^2 + 4)(12x)) / (6x^2 + 4)^4
= (8(36x^4 + 48x^2 + 16) - 16x(72x^3 + 48x)) / (6x^2 + 4)^4
= (288x^4 + 384x^2 + 128 - 1152x^4 - 768x^2) / (6x^2 + 4)^4
= (-864x^4 - 384x^2 + 128) / (6x^2 + 4)^4
Now, set the second derivative equal to zero and solve for x:
(-864x^4 - 384x^2 + 128) / (6x^2 + 4)^4 = 0
Since the numerator can never be equal to zero, we can focus on the denominator:
(6x^2 + 4)^4 = 0
The denominator is a perfect square, so it can never be equal to zero. Hence, there are no values of x where the second derivative is equal to zero.
Therefore, there are no inflection points for the given function f(x) = x^2 / (6x^2 + 4).