Question

In Exercises, find the point(s) of inflection of the graph of the function. $$ f(x)=x^{4}-18 x^{2}+5 $$

Short answer

The points of inflection for the function \(f(x)=x^4-18x^2+5\) are \((- \sqrt{3} , -22)\) and \(( \sqrt{3} , -22)\).

Step-by-step solution

Compute the first derivative

The first derivative of the function, \(f'(x)\), can be calculated by applying the power rules of differentiation. For \(f(x)=x^4-18x^2+5\), the first derivative, \(f'(x)\), is found as: \(f'(x) = 4x^3 - 36x\).

Compute the second derivative

To find the points of inflection, the second derivative will have to be found. The second derivative, \(f''(x)\), of the function can be found by differentiating the first derivative, \(f'(x)\). Applying the power rule, the second derivative \(f''(x)\) is found as: \(f''(x) = 12x^2 - 36\).

Set the second derivative equal to zero

To find the critical points, set the second derivative equal to zero and solve for \(x\). On solving \(12x^2 - 36 = 0\), we find \(x = \pm \sqrt{3}\) as the critical points.

Test the intervals around critical points

The intervals around the critical points, \(-\sqrt{3}\) and \(\sqrt{3}\), will be evaluated in the second derivative. If the sign of \(f''(x)\) changes on either side of the critical points, then they are points of inflection. After checking, it is observed that the function undergoes a change in concavity at both \(-\sqrt{3}\) and \(\sqrt{3}\), hence they are the points of inflection.

Find the coordinates of points of inflection

Substituting these points back into the original function will provide the corresponding y-values, giving the points of inflection as \((- \sqrt{3} , f(-\sqrt{3}))\) and \(( \sqrt{3} , f(\sqrt{3}))\). After calculating, we get the points of inflection to be \((- \sqrt{3} , -22)\) and \(( \sqrt{3} , -22)\).

Key concepts

Second Derivative Test

The second derivative test is a useful technique for identifying inflection points of a function. Inflection points occur where the concavity of a function changes, which means the second derivative changes its sign.

Here's how you use the second derivative test:

• First, find the second derivative, denoted as \(f''(x)\).
• Next, set \(f''(x)\) equal to zero and solve for \(x\) to find potential inflection points.
• Check the second derivative at intervals around these \(x\) values. A change in the sign (from positive to negative, or vice versa) indicates an inflection point.

In the exercise shared, after calculating \(f''(x) = 12x^2 - 36\) and equating it to zero, we find \(x = \pm \sqrt{3}\). Testing intervals around these points showed a change in concavity, confirming them as inflection points.

Concavity

Concavity describes the way a function curves. Understanding concavity is essential because it gives insights into the "shape" of the graph of the function.

A function can be:

• Concave up if its graph looks like a cup and the second derivative \(f''(x) > 0\).
• Concave down if it resembles a cap or an umbrella and \(f''(x) < 0\).

When a function transitions from concave up to concave down or vice versa at a certain point, this indicates an inflection point. In the given problem, the test around \(x = \pm \sqrt{3}\) constitutes the detection of changes in concavity. Before and after these points, the function's concavity switches, confirming them as inflection points.

Power Rule

The power rule is one of the basic rules for differentiation, critical in finding first and second derivatives. If \(f(x) = x^n\), the derivative \(f'(x)\) is determined by the formula \(n\cdot x^{n-1}\).

This rule simplifies computing derivatives of polynomial terms. Using the power rule:

• The first derivative of \(f(x) = x^4 - 18x^2 + 5\) becomes \(f'(x) = 4x^3 - 36x\), derived from applying \((4\cdot x^{4-1}) + (-18\cdot x^{2-1})\).
• The second derivative \(f''(x) = 12x^2 - 36\) follows, using the power rule again on the first derivative.

By efficiently breaking down these steps, identification of critical points and analysis of concavity transitions become manageable, making the power rule an essential differentiation tool.