Problem 222 - Calculus Volume 1

Question

For the following exercises, determine a. intervals where \(f\) is increasing or decreasing and b. local minima and maxima of \(f\) . $$ f(x)=x^{2}+\cos x $$

Step-by-Step Solution

Answer

Find critical points of \(f'(x)=2x-\sin x\) and test intervals for increase or decrease.

Step 1: Find the First Derivative

To find the intervals where the function is increasing or decreasing, we need to calculate the first derivative of \(f(x) = x^2 + \cos x\). The derivative is given by \(f'(x) = 2x - \sin x\). This derivative will help us determine the critical points and the intervals of increase or decrease.

Step 2: Find Critical Points

Critical points occur where the first derivative is zero or undefined. Set \(f'(x) = 0\) to find these points: \(2x - \sin x = 0\) which simplifies to \(2x = \sin x\). Solve for \(x\) using numerical methods or graphing, as the equation is transcendental and does not have a simple algebraic solution.

Step 3: Test Intervals for Increase or Decrease

With the critical points found in Step 2, choose test points in the intervals defined by these critical points and evaluate the sign of \(f'(x)\):- If \(f'(x) > 0\), \(f(x)\) is increasing.- If \(f'(x) < 0\), \(f(x)\) is decreasing.

Step 4: Determine Local Minima and Maxima

Use the First Derivative Test to classify the critical points found:- If \(f'(x)\) changes from positive to negative at a critical point, it's a local maximum.- If \(f'(x)\) changes from negative to positive, it's a local minimum.

Key Concepts

Critical Points

Critical points are crucial in understanding the behavior of a function. They are determined by finding where the first derivative of the function equals zero or is undefined. In simple terms, they are the points where the function changes direction.
For the function in our exercise, the derivative is given by \( f'(x) = 2x - \sin x \). To find critical points, we set this derivative equal to zero: \( 2x - \sin x = 0 \).
This simplifies to \( 2x = \sin x \). To solve this equation, we often need to use numerical methods or graph the equation, as it doesn't have a straightforward algebraic solution.
Finding these points helps in identifying sections of the graph where the function may have peaks (maxima) or troughs (minima).

Derivative

The derivative is a fundamental concept in calculus, representing the rate at which a function is changing at any given point. When you hear 'slope of a tangent' or 'instantaneous rate of change,' it directly refers to the derivative.
In our problem, the derivative of the function \( f(x) = x^2 + \cos x \) is \( f'(x) = 2x - \sin x \). This expression tells us how the function's slope changes across different values of \(x\).
Derivatives allow us to locate critical points, determine intervals of increase or decrease, and even identify potential maxima and minima. They are powerful tools for analyzing the behavior of functions.

Increasing and Decreasing Intervals

To find where a function is increasing or decreasing, we analyze the sign of the first derivative.
For our exercise, we use \( f'(x) = 2x - \sin x \). By examining this expression:

• If \( f'(x) > 0 \), the function is increasing in that interval.
• If \( f'(x) < 0 \), the function is decreasing in that interval.

Once we have identified the critical points by solving \( 2x - \sin x = 0 \), these points divide the number line into intervals. We pick test points in each interval to check the sign of \( f'(x) \) and thus determine whether the function is increasing or decreasing.
This method is known as the first derivative test and helps visualize how a function behaves across its domain.

Local Minima and Maxima

Local minima and maxima are points where a function reaches its lowest or highest value in some neighborhood. These points are critical because they inform us about potential points of interest in a function's graph.
To determine local minima or maxima using calculus, we utilize the critical points found previously. For \( f(x) = x^2 + \cos x \):

• If \( f'(x) \) changes from positive to negative as you pass through a critical point, that point is a local maximum.
• If \( f'(x) \) changes from negative to positive, that point is a local minimum.

This analysis tells us where the function may peak or dip, aiding in a deeper understanding of the function's shape. Remember, finding local minima and maxima provides a way to describe the terrain of the graph, highlighting high and low points without considering the function's behavior at infinite limits.