At what value of x does the graph of y=1/x^2 - 1/x^3 have a point of inflection?
To determine the value of x at which the graph has a point of inflection, we need to find the second derivative of the function and set it equal to zero.
Given the function y = 1/x^2 - 1/x^3, let's find its first derivative:
y' = d/dx (1/x^2) - d/dx (1/x^3)
Using the power rule for differentiation, we have:
y' = (-2/x^3) - (-3/x^4)
= -2/x^3 + 3/x^4
Now, let's find the second derivative by differentiating y' with respect to x:
y" = d/dx (-2/x^3 + 3/x^4)
Using the power rule again, we have:
y" = 6/x^4 - 12/x^5
To find the value of x at which the graph has a point of inflection, we need to set y" equal to zero and solve for x:
6/x^4 - 12/x^5 = 0
To simplify the equation, let's multiply both sides by x^5, resulting in:
6x - 12 = 0
Rearranging the equation, we have:
6x = 12
Dividing both sides by 6, we find:
x = 2
Therefore, the graph of y = 1/x^2 - 1/x^3 has a point of inflection at x = 2.
To find the value of x at which the graph of the equation y = 1/x^2 - 1/x^3 has a point of inflection, we need to find the second derivative and set it equal to zero.
1. First, let's find the first derivative of y with respect to x:
y = 1/x^2 - 1/x^3
dy/dx = d/dx(1/x^2) - d/dx(1/x^3)
2. Using the power rule, the derivatives are:
dy/dx = -2/x^3 + 3/x^4
3. Now, let's find the second derivative by differentiating the first derivative:
d^2y/dx^2 = d/dx(-2/x^3 + 3/x^4)
4. Using the power rule again, the second derivative is:
d^2y/dx^2 = 6/x^4 - 12/x^5
5. Setting the second derivative equal to zero:
6/x^4 - 12/x^5 = 0
6. To solve for x, multiply both sides of the equation by x^5:
6x - 12 = 0
7. Simplify the equation:
6x = 12
8. Divide both sides by 6:
x = 2
Therefore, the graph of y = 1/x^2 - 1/x^3 has a point of inflection at x = 2.