find the point(s) of inflection on the function f(x)= 10x + 10/x
find the point(s) of inflection on the function f(x)= 10/x + 10/x^2
f' = 10 - 10/x^2
f" = 0 - 10 [ -2/x^3] = 20/x^3
I do not see anyplace where the curvature passes through zero
except at x = +/- infinity
To find the point(s) of inflection on the function f(x) = 10x + 10/x, we need to follow these steps:
Step 1: Find the second derivative of the function.
Step 2: Set the second derivative equal to zero.
Step 3: Solve the equation to find the values of x.
Step 4: Test the sign of the second derivative to confirm if the points are inflection points.
Let's go through each step in detail:
Step 1: Find the second derivative of the function.
Given, f(x) = 10x + 10/x
To find the second derivative, we differentiate the function twice. The first derivative is found using the power rule and quotient rule:
f'(x) = d/dx (10x) + d/dx (10/x)
= 10 - 10/x^2
Now, differentiate f'(x) to find the second derivative:
f''(x) = d/dx (10 - 10/x^2)
= 0 + 20/x^3
= 20/x^3
Step 2: Set the second derivative equal to zero.
To find the points of inflection, we set f''(x) = 20/x^3 equal to zero:
20/x^3 = 0
Step 3: Solve the equation to find the values of x.
To solve 20/x^3 = 0, we need to find the values of x that make the expression equal to zero. Since it is a fraction, the numerator must be zero:
20 = 0
This equation has no solutions, so there are no x-values where f''(x) = 0 or points of inflection for this function.
Step 4: Test the sign of the second derivative to confirm if the points are inflection points.
Since there are no values of x where f''(x) = 0, we cannot perform this step. However, had there been roots in Step 3, we would have tested the sign of the second derivative to determine if the points are inflection points.
In summary, the function f(x) = 10x + 10/x does not have any points of inflection as there are no values of x where the second derivative is equal to zero.