Question

Locate the critical points of the following functions and use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. $$h(x)=(x+a)^{4} ; a \text { constant }$$

Short answer

Question: Determine if the critical point of the function $$h(x) = (x + a)^4$$ is a local maximum, local minimum, or inconclusive based on the Second Derivative Test. Answer: The Second Derivative Test is inconclusive for the critical point of the function $$h(x) = (x + a)^4$$.

Step-by-step solution

Find the first derivative of the function

To find the first derivative, use the power rule: $$h'(x) = \frac{d}{dx} (x + a)^4$$ $$h'(x) = 4(x + a)^3$$

Find the critical points

To find the critical points, set the first derivative equal to 0 and solve for x: $$4(x + a)^3 = 0$$ Solving for x gives us the only critical point: $$x = -a$$

Find the second derivative of the function

To find the second derivative, differentiate the first derivative: $$h''(x)= \frac{d}{dx}(4(x + a)^3)$$ $$h''(x)= 12(x + a)^2$$

Apply the Second Derivative Test

To apply the Second Derivative Test, substitute the critical point x = -a into the second derivative: $$h''(-a) = 12((-a) + a)^2$$ $$h''(-a) = 12(0)^2$$ $$h''(-a) = 0$$ Since the second derivative at the critical point is equal to 0, the Second Derivative Test is inconclusive, and we cannot determine if the critical point corresponds to a local maximum or local minimum.

Key concepts

Critical Points

In the quest to understanding the landscape of a function, critical points serve as our key markers. These points are where the function's first derivative is either zero or undefined, indicating potential peaks, valleys, or plateaus. In the exercise, we identified the critical point of the function by setting the first derivative equal to zero, which gave us a lone sentinel at the point where \( x = -a \). But the journey doesn't end here; knowing a critical point's location merely sets the stage. The true nature of this point—whether it's a highland, lowland, or just a flat terrain—requires further investigation with derivative tests.

First Derivative

The first derivative of a function, denoted as \( f'(x) \), is the backbone of calculus when it comes to analyzing the function's rate of change. Simply put, it tells us how the function's output changes as its input changes. When we calculated the first derivative for our function \( h'(x) = 4(x + a)^3 \), we've actually mapped out the slope of the tangent line at any point along the function. At \( x = -a \), our first derivative became zero, indicating a possible turning point. Understanding how to find and interpret the first derivative is fundamental for delving into the function's behavior.

Power Rule

Simplifying the process of differentiation, the power rule is a swift tool in our mathematical arsenal. It states that if you have a function of the form \( x^n \), its derivative is \( nx^{n-1} \). This rule saves us time and reduces complexity, especially with polynomials. In our example, applying the power rule transformed the function \( h(x) = (x + a)^4 \), into its derivative \( h'(x) = 4(x + a)^3 \) with elegance and efficiency. It's a straightforward concept but a pillar of calculus nevertheless.

Local Maximum and Minimum

Local maximums and minimums are the highs and lows of a function within a certain interval, resembling peaks and troughs of a mountain range. They provide us with significant insights into where the function reaches its extreme values. However, locating these points is not always a clear-cut process. The Second Derivative Test we aimed to use, which involves substituting the critical point into the function's second derivative, would usually tell us whether we have a concave up (indicating a minimum) or concave down (indicating a maximum) on the graph. But sometimes, like with our function \( h''(-a) = 0 \), the test is inconclusive, leaving us in uncertainty about the function's local extremities at that critical point.