Question

Local extreme points and inflection points Suppose that \(f\) has two continuous derivatives at \(a\) a. Show that if \(f\) has a local maximum at \(a,\) then the Taylor polynomial \(p_{2}\) centered at \(a\) also has a local maximum at \(a\) b. Show that if \(f\) has a local minimum at \(a,\) then the Taylor polynomial \(p_{2}\) centered at \(a\) also has a local minimum at \(a\) c. Is it true that if \(f\) has an inflection point at \(a,\) then the Taylor polynomial \(p_{2}\) centered at \(a\) also has an inflection point at \(a ?\) d. Are the converses to parts (a) and (b) true? If \(p_{2}\) has a local extreme point at \(a,\) does \(f\) have the same type of point at \(a ?\)

Short answer

Answer: No, a second-degree Taylor polynomial cannot have an inflection point at a point \(a\) even if the original function has an inflection point at the same point. This is because the second-degree Taylor polynomial has a fixed sign of its second derivative, while the original function's second derivative changes sign at the inflection point.

Step-by-step solution

Part a: Show that if \(f\) has a local maximum at \(a,\) then the Taylor polynomial \(p_{2}\) centered at \(a\) also has a local maximum at \(a\)

If \(f\) has a local maximum at \(a\), then \(f'(a) = 0\) . The second-degree Taylor polynomial of \(f\) centered at \(a\) is given by \(p_2(x) = f(a)+f'(a)(x-a)+\frac{1}{2}f''(a)(x-a)^2\). At \(x = a\), we have: \(p_2(a) = f(a) + f'(a)(a-a) + \frac{1}{2} f''(a)(a-a)^2 = f(a)\). Since \(f'(a) = 0\), we can rewrite the Taylor polynomial as \(p_2(x) = f(a) + \frac{1}{2}f''(a)(x-a)^2\). Now, since \(f'(a)=0\) and \(f\) has a local maximum at \(a\) , we have \(f''(a)\le0\). Thus, \(p_2(x)\) is a parabola with the vertex at the local maximum \(a\) and downward-facing.

Part b: Show that if \(f\) has a local minimum at \(a,\) then the Taylor polynomial \(p_{2}\) centered at \(a\) also has a local minimum at \(a\)

The argument here is similar to that in part a. If \(f\) has a local minimum at a, then \(f'(a) = 0\). Using the same second-degree Taylor polynomial expression as before, we can rewrite it as \(p_2(x) = f(a) + \frac{1}{2}f''(a)(x-a)^2\). Now, since \(f'(a)=0\) and \(f\) has a local minimum at \(a\) , we have \(f''(a)\ge0\). Thus, \(p_2(x)\) is a parabola with the vertex at the local minimum \(a\) and upward-facing.

Part c: Is it true that if \(f\) has an inflection point at \(a,\) then the Taylor polynomial \(p_{2}\) centered at \(a\) also has an inflection point at \(a\)

No, this is not necessarily true. A function \(f\) has an inflection point at \(a\) if its second derivative \(f''(a)\) changes sign at \(a\). However, the second-degree Taylor polynomial \(p_2(x) = f(a)+f'(a)(x-a)+\frac{1}{2}f''(a)(x-a)^2\) has a fixed sign of its second derivative, which is \(f''(a)\). As a result, the Taylor polynomial cannot have an inflection point at \(a\) even if the original function does.

Part d: Are the converses to parts (a) and (b) true? If \(p_{2}\) has a local extreme point at \(a,\) does \(f\) have the same type of point at \(a?\).

The converses for parts (a) and (b) are not always true. If the second-degree Taylor polynomial \(p_2(x)\) has a local maximum (minimum) at \(a\), it is true that \(f'(a) = 0\) and \(f''(a)\le0\) (or \(f''(a)\ge0\)). However, this does not guarantee that \(f(x)\) has a local maximum (minimum) at \(a\). It is possible for higher-order derivatives to affect the behavior of the function in the neighborhood of \(a\), and therefore \(f(x)\) might not have a local extreme point at \(a\) even if the second-degree Taylor polynomial does.

Key concepts

Local Maxima and Minima

Understanding local maxima and minima is crucial in calculus to determine the peaks and valleys of a function. At a local maximum, the function reaches a peak point in its neighborhood, while at a local minimum, it dips. For a function \(f(x)\), if \(f\) has a local maximum at some point \(a\), then a comparison of its nearby values confirms that no greater nearby value exists. Similarly, for a local minimum, \(f(a)\) is the smallest compared to its surroundings.

• If the first derivative, \(f'(a)\), equals zero, the function may either have a maximum or minimum at \(a\).
• If the second derivative, \(f''(a)\), is less than zero, \(f(a)\) indicates a local maximum.
• If \(f''(a)\) is greater than zero, \(f(a)\) indicates a local minimum.
• These insights are reflected in the Taylor polynomial \(p_2(x)\), driving its form based on the sign of \(f''(a)\).

Inflection Points

Inflection points offer vital information about how curves change direction. An inflection point occurs where a function changes concavity. In simpler terms, it is where a curve turns from being cup-shaped (concave up) to cap-shaped (concave down), or vice versa.

• For a function \(f(x)\), meanwhile, these points are identified where the second derivative, \(f''(a)\), transitions in sign. This means that \(f''(x)\) must pass through zero or become undefined at an inflection point.
• However, the second-degree Taylor polynomial \(p_2(x)\) doesn't adapt its shape through inflection points, as its second derivative remains constant throughout its domain.

It's important to note because the lack of sign change in the polynomial's second derivative \(\frac{1}{2}f''(a)\) prohibits it from having inflection points.

Second Derivative Test

The second derivative test is a handy tool to pinpoint where the local maxima and minima occur along a curve. When \(f'(a) = 0\), additional information from the second derivative \(f''(a)\) assists in clearly determining the nature of \(a\).
For instance:

• \(f''(a) > 0\) implies \(a\) is a local minimum.
• \(f''(a) < 0\) implies \(a\) is a local maximum.
• If \(f''(a) = 0\), the test is inconclusive. Higher derivatives might be needed to assess the behavior of \(f(x)\) around \(a\).

The second-degree Taylor polynomial helps illustrate this by its curvature, though it doesn't always correlate exactly with the function's higher-order derivatives.

Converses in Calculus

Discussing converses entails examining situations where the outcomes are flipped. Meaning, if the Taylor polynomial \(p_2(x)\) shows a local extreme at \(a\), does \(f(x)\) share this characteristic?

• The fact that \(p_2(x)\) exhibits a behavior doesn't ensure \(f(x)\) will display the same, due to the complexity and possible dominance of higher-order derivatives in \(f(x)\).
• Conversations around converses remind us that initial observations (like those from \(p_2(x)\)) might mask broader truths hidden in \(f(x)'s\) higher derivatives.