Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ f(x)=x^{4}-2 x^{2}+2 ; I=[-2,2] $$

Step-by-Step Solution

Answer

Maximum value: 10; Minimum value: 1.

Step 1: Find the Derivative

To find the critical points, we first need to find the derivative of the function. Given \( f(x) = x^4 - 2x^2 + 2 \), the derivative \( f'(x) \) is obtained by differentiating each term with respect to \( x \):\[f'(x) = 4x^3 - 4x.\]

Step 2: Set Derivative Equal to Zero

To find the critical points, set the derivative equal to zero:\[4x^3 - 4x = 0.\]Factor the equation:\[4x(x^2 - 1) = 0.\]

Step 3: Solve for Critical Points

Solve the factored equation:- From \( 4x = 0 \), we get \( x = 0 \).- From \( x^2 - 1 = 0 \), we get \( x^2 = 1 \), so \( x = \pm1 \).Hence, the critical points are \( x = -1, 0, 1 \).

Step 4: Evaluate Function at Critical Points and Endpoints

Evaluate \( f(x) \) at the critical points and the endpoints of the interval \([-2, 2]\):- \( f(-2) = (-2)^4 - 2(-2)^2 + 2 = 16 - 8 + 2 = 10 \).- \( f(-1) = (-1)^4 - 2(-1)^2 + 2 = 1 - 2 + 2 = 1 \).- \( f(0) = 0^4 - 2(0)^2 + 2 = 2 \).- \( f(1) = 1^4 - 2(1)^2 + 2 = 1 - 2 + 2 = 1 \).- \( f(2) = 2^4 - 2(2)^2 + 2 = 16 - 8 + 2 = 10 \).

Step 5: Determine Maximum and Minimum Values

Compare the values found in Step 4. The maximum value on the interval is \( f(x) = 10 \) (occurring at \( x = -2 \) and \( x = 2 \)), and the minimum value is \( f(x) = 1 \) (occurring at \( x = -1 \) and \( x = 1 \)).

Key Concepts

Critical Points in Calculus Optimization

Critical points are essential in calculus optimization as they help us identify where a function's greatest extremes might occur within a given interval. These points are where the function's derivative is either zero or undefined. For the function \( f(x)=x^4-2x^2+2 \), critical points are found by first calculating its derivative, \( f'(x) = 4x^3 - 4x \). By setting the derivative equal to zero, we obtain an equation that can be solved to determine the critical points:

• \(4x = 0\) leads to \(x = 0\).
• \(x^2 - 1 = 0\) results in \(x = \pm 1\).

Thus, the critical points for this function are \(x = -1, 0, 1\). These are the points at which we examine the behavior of the function to identify any potential maximum or minimum values within the given interval.

Understanding Derivatives

Derivatives are a fundamental concept in calculus that measure how a function changes as its input changes. They indicate the slope of the tangent line to the function at any given point, revealing the rate at which the function's value increases or decreases. To find the derivative of \( f(x)=x^4-2x^2+2 \), we differentiate each term:

• The derivative of \(x^4\) is \(4x^3\).
• The derivative of \(-2x^2\) is \(-4x\).
• The derivative of the constant \(2\) is \(0\).

This gives us the derivative \(f'(x) = 4x^3 - 4x\). Knowing the derivative is crucial for identifying critical points and understanding the function's behavior concerning its increase or decrease at various intervals.

Exploring Maximum and Minimum Values

After identifying critical points, we need to evaluate the function at these points and endpoints of the interval to find any maximum and minimum values. For functions like \( f(x)=x^4-2x^2+2 \) on the interval \([-2, 2]\), this involves three main steps:1. **Evaluate at Critical Points**: Substitute the critical points \(x = -1, 0, 1\) into the function to find corresponding function values. 2. **Evaluate at Endpoints**: Compute the function values at the interval's endpoints, \(x = -2\) and \(x = 2\).3. **Determine Extremes**: Compare all calculated values.

• The maximum value \(f(x) = 10\) occurs at \(x = -2\) and \(x = 2\).
• The minimum value \(f(x) = 1\) occurs at \(x = -1\) and \(x = 1\).

This methodical approach ensures that we have identified the function's highest and lowest points within the specified interval.

Interval Analysis in Calculus

Interval analysis in calculus is about examining how a function behaves over a particular range of its input values, called an interval. Within this interval, we analyze the function to determine if it has any critical points and to find its maximum and minimum values.For \( f(x)=x^4-2x^2+2 \) on the interval \([-2, 2]\), we:1. **Find Critical Points**: Use the derivative to ascertain where the function's rate of change is zero. This helps locate potential extremities.2. **Evaluate the Function**: Calculate the function value at critical points and at the interval's boundaries.3. **Compare Values**: Compare all these values to ascertain the function’s maximum and minimum within the interval.Interval analysis guides us in understanding where a function achieves significant values and how it changes over a specified range, providing insight into its overall behavior and key characteristics.