Question

In Exercises \(23-28\) , find (a) the local extrema, (b) the intervals on which the function is increasing, and (c) the intervals on which the function is decreasing. $$f(x)=x^{3}-2 x-2 \cos x$$

Short answer

To find the local extrema and intervals of increase or decrease, you have to follow the steps of differentiating the function, finding the critical points, and comparing the intervals. Pay attention to the signs of the first and second derivatives, as they reveal information about the rate of change and 'bends' in the function, respectively.

Step-by-step solution

Derive the Function

First thing to do is differentiate the function \(f(x)=x^{3}-2 x-2 \cos x\). The derivative, \(f'(x)\), will help us understand the rate of change of the function - where it's increasing or decreasing and where it reaches extrema.

Calculate Critical Points

Next, find the critical points, these exist where the derivative is zero or undefined. Solve \(f'(x) = 0\). The solutions are the critical points.

Local Extrema Identification

Along these critical points, we can identify local extrema. This can be done by using the second derivative test. Find the second derivative \(f''(x)\). Plugging the critical points into it, if the value is positive at \(x_{c}\), then we have a local minimum at \(x_{c}\). If it's negative, we have a maximum at that point.

Interval Comparison

Now, we compare the critical points. For every two consecutive critical points, plug any number that falls within this interval into the derivative. If \(f'(x)>0\), the function is increasing in that interval. If \(f'(x)<0\), it's decreasing there.

Key concepts

Critical Points in Calculus

A critical point in calculus is a spot on the graph of a function where the derivative is either zero or undefined. These points are crucial for understanding the behavior of functions because they can indicate where local extrema occur – either a peak (maximum) or a trough (minimum).

When you come across a problem that asks you to find the local extrema of a function like
\[f(x) = x^3 - 2x - 2\cos(x)\],
the first step is to take its derivative to get
\[f'(x)\].

This derivative represents the slope of the function at any given point. Wherever this slope is zero, the function either levels off or changes direction, signaling a potential extreme point. Let's say if we find
\[f'(x) = 0\],
the corresponding x-values are the critical points we are interested in. These critical points are essential to determine where the function can reach its highest or lowest values, within certain intervals, and represent potential turning points in the behavior of the function.

To fully grasp the concept, it's extremely useful to spend some time plotting a few functions and their derivatives, identifying the zeros of the derivative, and observing how these points relate to the peaks and troughs on the original function's graph.

Derivative Test for Local Extrema

Once critical points are identified, the derivative test comes into play to clarify whether these points are local maxima, minima, or neither. This is where the second derivative of the original function
\[f''(x)\]
becomes particularly valuable.

The sign of the second derivative at a critical point determines the nature of the extremum:

• If
  \[f''(x_c) > 0\],
  the function has a local minimum at \(x_c\).
• If
  \[f''(x_c) < 0\],
  there is a local maximum at that point.
• If
  \[f''(x_c) = 0\],
  the test is inconclusive, and further investigation is needed.

Let's walk through an example. For the given function
\[f(x) = x^3 - 2x - 2\cos(x)\],after finding its critical points, we would then compute the second derivative and evaluate it at those critical points. This tells us whether the graph is concave up (cup-like) or concave down (cap-like) at those points, and subsequently, if the critical points are associated with local maxima or minima.

Intervals of Increase and Decrease

The final step after identifying the critical points and testing them for local extrema is to determine on which intervals the function is increasing or decreasing. This determination is based on the sign of the first derivative of the function over various intervals.

By selecting test points between the critical points and plugging them into the first derivative
\[f'(x)\],
you can ascertain the function's behavior. If the first derivative is positive between two critical points, the function is increasing on that interval. Conversely, if the derivative is negative, the function is decreasing.

Let's relate this to our function
\[f(x) = x^3 - 2x - 2\cos(x)\].
Suppose we found some critical points; now, by choosing a number in-between each pair of critical points and evaluating the first derivative at that number, we would see if the function slopes upwards or downwards in that segment.

• If
  \[f'(x) > 0\],
  it suggests the function is heading upward as you move from left to right.
• If
  \[f'(x) < 0\],
  the function is on a decline in that same direction.

The knowledge of these intervals is exceptionally beneficial, as it provides a better insight into the overall trend of the function, beyond just knowing where the peaks and valleys lie.