Question

In Exercises \(1-6,\) use the First Derivative Test to determine the local extreme values of the function, and identify any absolute extrema. Support your answers graphically. $$y=2 x^{4}-4 x^{2}+1$$

Short answer

The function \(y=2 x^{4}-4 x^{2}+1\) has local minima at \(x=-1\) and \(x=1\), and a local maximum at \(x=0\). These are also the absolute extrema, with absolute minima \(y=1\) at \(x=-1\) and \(x=1\), and absolute maximum \(y=1\) at \(x=0\).

Step-by-step solution

Find the First derivative

First, find the derivative of the function \(y=2 x^{4}-4 x^{2}+1\). Use the power rule for differentiation, which states that the derivative of \(x^n\) is \(nx^{n-1}\), we get \(y'=8x^{3}-8x\).

Find the critical points

Set the first derivative to zero and solve for \(x\). So the equation is \(8x^{3}-8x=0\). Factoring out \(8x\), we get \(8x(x^{2}-1)=0\). Solving this gives \(x=0\), \(x=1\), and \(x=-1\). These are the critical points.

Apply First Derivative Test

To determine whether these critical points are local maxima or minima, plug some values into the derivative that lie in the intervals \(x<-1\), \(-11\). If \(y'\) changes sign from + to - as \(x\) increases, the function has a local maximum at that point. If \(y'\) changes sign from - to +, the function has a local minimum at that point. Doing so, it is determined that \(x=-1\) and \(x=1\) are local minima and \(x=0\) is a local maximum.

Find the absolute extrema

The absolute extrema are the highest and lowest points on the graph, which means evaluating the function at the critical points and at the endpoints of its domain. However, since the given function is a polynomial, it has no end points, its domain is all real numbers. Evaluating the function at the critical points, we get the following \(y\)-values: \(y(-1)=y(1)=1\) and \(y(0)=1\). So the only extrema are absolute minima at \(x=-1\), \(x=1\), and absolute maximum at \(x=0\).

Key concepts

Critical Points

Understanding critical points is crucial when analyzing the behavior of functions. These are the points on a graph where the function's first derivative is either zero or undefined. In simpler terms, they are points where the function's graph either has a horizontal tangent or no tangent at all.

Finding critical points is the first step towards identifying possible locations for local extrema. As seen in the given exercise, we determine the critical points for the function \(y=2x^4-4x^2+1\) by setting its derivative \(y'=8x^3-8x\) equal to zero and solving for \(x\). In this case, we obtain the critical points at \(x=0\), \(x=1\), and \(x=-1\).

To practically apply this concept, you should always remember that a critical point is not necessarily a point of extremum. It's more like a candidate that warrants further investigation through tools like the First Derivative Test.

Local Extrema

Local extrema refer to the highest or lowest values a function takes within a certain interval around a point. These are essentially the peaks and valleys of a graph. A local maximum is the peak point where the function's value is the highest in the immediate vicinity, while a local minimum is like a valley bottom where the function's value is the lowest nearby.

The First Derivative Test helps us find these local extrema by analyzing the change in sign of the first derivative around the critical points. If \(y'\) changes from positive to negative, this indicates a local maximum. Conversely, if \(y'\) changes from negative to positive, this indicates a local minimum. Following the steps from the exercise solution, we concluded that \(x=-1\) and \(x=1\) are points of local minima because of this sign change in the derivative, while \(x=0\) is a point of local maximum.

Remember, local extrema are context-sensitive; they are the 'highs' and 'lows' relative to their immediate surroundings on the graph, and not necessarily the absolute highest or lowest points of the entire function's domain.

Absolute Extrema

While local extrema are concerned with their immediate environment, absolute extrema are the function's highest and lowest points over its entire domain. When we talk about absolute maximum and minimum, we are referring to the entire graph of the function, not just a section of it.

The search for absolute extrema involves evaluating the function at the critical points as well as the boundaries of the domain. However, in our case with a polynomial function, there are no boundaries to consider because polynomials are defined for all real numbers.

Applying this to our exercise, after evaluating the polynomial at the critical points found previously (\(x=0\), \(x=1\), and \(x=-1\)), we determine the absolute maximum and minimum values of the function. Since polynomials tend to infinity in either direction, we focus only on the critical points, identifying \(x=0\) as the absolute maximum and \(x=-1\) and \(x=1\) as the points of absolute minima. This comprehension of absolute extrema is essential as it helps to completely understand the overall behavior of functions.