Problem 6 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=-2 x^{3}+3 x^{2}+12 x+5\) on \([-3,4]\)

Step-by-Step Solution

Answer

All critical points, their classifications, and the absolute maximum and/or minimum of the function will be deduced after completion of the above steps.

Step 1: Find the derivative and critical points

Differentiate the function \(f(x)=-2x^{3}+3x^{2}+12x+5\) to find \(f'(x)\). Now equate \(f'(x)\) to zero to find the critical points.

Step 2: Classify the critical points

Substitute each critical point into the second derivative of the function \(f''(x)\). If \(f''(x)\) is positive at a critical point, then the function has a local minimum there. If \(f''(x)\) is negative, then the function has a local maximum. If \(f''(x)=0\) at a critical point or is undefined, then test the point using the First or Second Derivative Test to determine its nature.

Step 3: Find the absolute maximum and minimum

Evaluate the function \(f(x)\) at the endpoints of the interval \([-3,4]\) and at the critical points. The largest of these values is the absolute maximum and the smallest is the absolute minimum of the function on the interval.

Key Concepts

Derivative Test

Understanding derivatives is crucial in calculus, especially when finding critical points for functions. The derivative of a function, often written as \(f'(x)\), represents the rate at which the function changes at a certain point. To find critical points, which can be potential locations for maxima, minima, or points of inflection, you first need to derive the function and set \(f'(x) = 0\). This gives you the x-values where the slope of the function is zero, indicating these critical spots. Make sure to also check if \(f'(x)\) is undefined at any point within the interval, as this may indicate additional critical points.

• Differentiate the function to get \(f'(x)\).
• Set \(f'(x)\) to zero and solve for \(x\).
• Check for any undefined points within the specified interval.

Once you have identified the critical points, the next step involves classifying these points to understand the function's behavior at those points.

Local Maximum and Minimum

When analyzing functions, finding and classifying local maxima and minima is a key part of understanding the function's graph. After identifying critical points using the derivative test, we need to determine if each critical point is a local maximum, minimum, or possibly neither. This is where the second derivative test becomes useful. For a function \(f(x)\):

• If \(f''(x) > 0\) at a critical point, the function has a local minimum there.
• If \(f''(x) < 0\) at a critical point, the function has a local maximum.
• If \(f''(x) = 0\), the point might be an inflection point, and further testing is needed.

Always remember that local maxima and minima are only considered within a neighborhood of the function, not over the entire interval. This means a local maximum is the highest point in the vicinity, and a local minimum is the lowest, compared to nearby points.

Absolute Maximum and Minimum

Determining the absolute maximum and minimum of a function over a specified interval involves evaluating the function at various key points. These points include the endpoints of the interval and any critical points found within the interval. To find these absolute values:

• Calculate the function value at each endpoint of the interval.
• Evaluate the function at each critical point.
• Compare all these values.

The highest function value from your evaluations is the absolute maximum, and the lowest is the absolute minimum within the given interval. These absolute points provide the highest and lowest possible points the function can reach over the interval, ensuring a comprehensive understanding of overall behavior.