Problem 71 - Calculus Early Transcendentals: Pearson New International Edition

Question

Draw the graphs of \(f(x)=x^{3}-4 x^{2}+3\) and its derivative \(f^{\prime}(x)\) on the interval \([-2,5]\) using the same axes. (a) Where on this interval is \(f^{\prime}(x)<0\) ? (b) Where on this interval is \(f(x)\) decreasing? (c) Make a conjecture. Experiment with other intervals and other functions to support this conjecture.

Step-by-Step Solution

Answer

(a) \(0 < x < \frac{8}{3}\). (b) Same interval. (c) A function decreases where its derivative is negative.

Step 1: Find the Derivative

To find the derivative of the function \(f(x) = x^3 - 4x^2 + 3\), we use the power rule. The derivative, \(f'(x)\), is found by differentiating each term: \( \frac{d}{dx} x^3 = 3x^2 \) and \( \frac{d}{dx} (-4x^2) = -8x \). The constant term \(3\) disappears when differentiated. Thus, \(f'(x) = 3x^2 - 8x\).

Step 2: Determine Where the Derivative is Negative

A function \(f(x)\) is decreasing where its derivative \(f'(x) < 0\). We set the inequality \(3x^2 - 8x < 0\). Factoring out \(x\) gives \(x(3x - 8) < 0\). The critical points are at \(x = 0\) and \(x = \frac{8}{3}\). By testing intervals, we determine \(f'(x) < 0\) on the interval (0, \(\frac{8}{3}\)).

Step 3: Analyze the Function's Behavior

Since \(f(x)\) is decreasing where \(f'(x)<0\), we conclude that \(f(x)\) is decreasing on the interval (0, \(\frac{8}{3}\)). The negative derivative on this interval implies the function is sloping downward.

Step 4: Visualize by Graphing

Graph \(f(x) = x^3 - 4x^2 + 3\) and its derivative \(f'(x) = 3x^2 - 8x\) on the closed interval \([-2, 5]\). The graph of \(f(x)\) should show it sloping downward on the interval \(0 < x < \frac{8}{3}\), which corresponds to the interval where \(f'(x)\) is below the x-axis (negative).

Step 5: Conjecture Formulation

From this exercise, we can conjecture that a function is decreasing on intervals where its derivative is negative. Experimenting with other polynomials or different intervals should support this conjecture that the slope of the tangent to a function being negative corresponds to the function itself decreasing.

Key Concepts

Derivative

In calculus, the derivative of a function is a fundamental concept used to understand how the function changes at any given point. For the function \( f(x) = x^3 - 4x^2 + 3 \), the derivative, noted as \( f'(x) \), provides us with the rate of change or the slope of the tangent line to the curve at each point along the curve.

The process to find the derivative of a polynomial involves the power rule, which states that for any term \( ax^n \), the derivative is \( n \cdot ax^{n-1} \). Applying this to \( f(x) \):

• The derivative of \( x^3 \) is \( 3x^2 \).
• The derivative of \(-4x^2 \) is \(-8x \).
• The constant \( 3 \) vanishes, as the derivative of a constant is zero.

This results in the derivative \( f'(x) = 3x^2 - 8x \).

Understanding derivatives allows us to determine where a function like \( f(x) \) is increasing, decreasing, or constant at different intervals.

Decreasing Function

A function is classified as decreasing when its output values drop as the input values increase. This is the essence of what we mean when we say a function is 'sloping downward'.

To determine when the given polynomial function \( f(x) = x^3 - 4x^2 + 3 \) is decreasing, we need to find where the derivative \( f'(x) = 3x^2 - 8x \) is less than zero:

• Start by solving \( 3x^2 - 8x < 0 \).
• Factoring gives \( x(3x - 8) < 0 \).
• The critical points, \( x = 0 \) and \( x = \frac{8}{3} \), split the number line into intervals.
• Testing intervals shows that \( f'(x) < 0 \) in \((0, \frac{8}{3})\).

This means \( f(x) \) is decreasing on the interval \( 0 < x < \frac{8}{3} \).

The function's graph will reflect this by sloping downwards in this region, indicating a decrease in the function's output values as the input increases.

Polynomial

A polynomial is a mathematical expression consisting of variables (often represented as \( x \)) raised to whole-number powers, with corresponding coefficients. The function \( f(x) = x^3 - 4x^2 + 3 \) is a cubic polynomial, as its highest degree term is \( x^3 \).

Polynomials are smooth and continuous, and their behavior is dictated by their highest power term. In this case, the cubic term \( x^3 \) exerts the most influence over the shape of the graph. Here's what different polynomial characteristics suggest:

• Degree: The highest power term (degree 3), indicating the graph has two inflection changes.
• Leading Coefficient: Positive, so the ends of the graph rise to infinity in opposite directions.
• Constant Term: \(3\), which gives the intercept on the y-axis.

Graphing polynomials like \( f(x) \) and understanding their degree and coefficients help predict interaction with axes, symmetry, turning points, and overall behavior.

Through graphing and analyzing polynomials, one can gain insights into the nature of functions and their derivatives, aiding in understanding concepts like increasing and decreasing intervals.