Question

Solve for the points of inflection on h’(x)=(x^2-2)/x

Answer 1

The second derivative of the function h'(x)=(x^2 - 2)/x is never zero, thus there are no points of inflection for this function.

Step-by-step explanation:

To solve for the points of inflection of the function h'(x) = (x^2 - 2) / x, we need to find the points where the second derivative, h''(x), changes sign. This indicates a change in the concavity of the graph of the function. First let's simplify h'(x). We get h'(x) = x - 2/x. Now, we take the second derivative, h''(x) = 1 + 2/x^2.

Now, let's find where h''(x) equals zero. Since 1 + 2/x^2 is never zero because 1 is always positive and 2/x^2 is always positive (except when x = 0, which is not in the domain of h'(x) since it causes division by zero), there are no x-values for which the second derivative equals zero, and hence, there are no points of inflection.

Answer 2

The function h(x) = (x^2 - 2) / x has no points of inflection because its second derivative, h''(x) = -4 / x^3, has no real solutions.

To find the points of inflection for a function h(x), you need to follow these steps:

Find the second derivative, h''(x).

Set h''(x) = 0 and solve for x.

Find the corresponding y values for the x values obtained in step 2.

Let's find the points of inflection for the given function h(x) = (x^2 - 2) / x.

Find the first derivative h'(x):

h'(x) = ((x * (2x) - (x^2 - 2) * 1) / x^2 = (2x^2 - x^2 + 2) / x^2 = (x^2 + 2) / x^2

Find the second derivative h''(x):

h''(x) = ((x^2 * 2 - (x^2 + 2) * 2x) / x^4 = (2x^2 - 2x^2 - 4x) / x^4 = -4x / x^4 = -4 / x^3

Set h''(x) = 0 and solve for x:

-4 / x^3 = 0

This equation has no solution because the numerator is a constant (-4) and cannot be equal to zero.

So, the given function h(x) = (x^2 - 2) / x does not have any points of inflection.