Problem 53 - Essential Calculus Early Transcendentals

Question

(a) If the function \(f(x)=x^{3}+a x^{2}+b x\) has the local minimum value \(-\frac{2}{9} \sqrt{3}\) at \(x=1 / \sqrt{3},\) what are the values of \(a\) and \(b ?\) (b) Which of the tangent lines to the curve in part (a) has the smallest slope?

Step-by-Step Solution

Answer

Find \( a \) and \( b \) by solving the system of equations derived from the local minimum condition. The smallest slope is \( f'(x) \) near its minimum.

Step 1: Determine conditions for local minimum

For the function \( f'(x) = 3x^2 + 2ax + b \) to have a local minimum at \( x = \frac{1}{\sqrt{3}} \), it must satisfy two conditions: 1) \( f'(\frac{1}{\sqrt{3}}) = 0 \) for the slope of the tangent to be zero, and 2) evaluate \( f''(x) = 6x + 2a \) to ensure it's positive at \( x = \frac{1}{\sqrt{3}} \).

Step 2: Apply derivative conditions

1) Calculate \( f'(x) = 3x^2 + 2ax + b \).Substitute \( x = \frac{1}{\sqrt{3}} \):\[ 3\left(\frac{1}{\sqrt{3}}\right)^2 + 2a\frac{1}{\sqrt{3}} + b = 0. \]Solve for \( a \) and \( b \) in terms of \( \frac{1}{\sqrt{3}} \).2) Ensure \( f''(\frac{1}{\sqrt{3}}) = 6\frac{1}{\sqrt{3}} + 2a > 0 \).Substitute this to find further relations.

Step 3: Substitute local minimum value into function

Evaluate \( f(x) = x^3 + ax^2 + bx \) at \( x = \frac{1}{\sqrt{3}} \).Set \( f\left(\frac{1}{\sqrt{3}}\right) = -\frac{2}{9}\sqrt{3} \), to solve:\[ \left(\frac{1}{\sqrt{3}}\right)^3 + a\left(\frac{1}{\sqrt{3}}\right)^2 + b\left(\frac{1}{\sqrt{3}}\right) = -\frac{2}{9}\sqrt{3}. \]

Step 4: Solve the system of equations

From the previous steps, you have two simultaneous equations involving \( a \) and \( b \): 1) \( \frac{1}{3} + \frac{2a}{3\sqrt{3}} + \frac{b}{\sqrt{3}} = 0 \)2) \( \frac{1}{3\sqrt{3}} + \frac{a}{3} + \frac{b}{\sqrt{3}} = -\frac{2}{9}\sqrt{3} \)Solve these equations to find \( a \) and \( b \).

Step 5: Identify the tangent line with smallest slope

To find the tangent line with the smallest slope, examine the derivative \( f'(x) = 3x^2 + 2ax + b \).Aggregate information about \( f''(x) = 6x + 2a \) to confirm that \( f'(x) \) has a minimum. Consider endpoints or behavior as \( x \to \infty \). If the local minimum of \( f(x) \) does not correspond to the smallest slope, analyze which value of \( x \) makes \( f'(x) \) smallest.

Key Concepts

Local Minimum

In calculus, a local minimum is a point on a graph where a function's value is lower than all other function values in its immediate vicinity. This concept is crucial for understanding how functions behave at certain input values, especially in optimization problems. To find a local minimum, we need to look at the derivative of the function.

• The first derivative of the function, denoted as \( f'(x) \), gives us the slope of the tangent line at any particular point \( x \).
• A local minimum occurs at a point \( x = c \) where \( f'(c) = 0 \), meaning the slope is horizontal at that point.
• Additionally, the second derivative \( f''(x) \) should be positive at that point for it to be a local minimum, confirming a concave up behavior.

In our problem, we verify that the function has a local minimum at a specific point by analyzing both the first and second derivatives.

Derivative Test

The Derivative Test is a fundamental tool for determining the local extremum of a function, like a local minimum or maximum. It involves the evaluation of the first and second derivatives of the function.To find a local minimum:

• First, set the first derivative \( f'(x) \) to zero. This condition identifies critical points, where the function's slope is either a local minimum, maximum or a saddle point.
• With the critical points calculated, check the second derivative \( f''(x) \). If \( f''(x) > 0 \) at the critical point, the point is a local minimum.

In the given exercise, we use the derivative test on the quadratic function to confirm the local minimum. This involves finding points where the slope of the tangent (given by \( f'(x) \)) turns zero and verifying the concavity with \( f''(x) \).

Tangent Line Slope

The slope of a tangent line to a curve at a point tells us how steep the curve is at that very location. The slope is essentially the value of the first derivative \( f'(x) \) of the function at a specific \( x \)-value.

• If the slope is positive, the function is increasing at that point. Conversely, if the slope is negative, the function is decreasing.
• When the slope is zero \( (f'(x) = 0) \), the tangent line is horizontal, hinting at a potential local maximum or minimum point.
• Finding the smallest slope for the tangent line involves examining \( f'(x) \) to detect where it reaches the lowest value possible.

For part (b) of the problem, finding the tangent line with the smallest slope means we look for the smallest value of \( f'(x) \), hence analyzing how \( f'(x) \) behaves especially outside the local minimum is important.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form \( f(x) = ax^2 + bx + c \). They graph into a parabolic shape, which can either open upward or downward depending on the value of \( a \).

• If \( a > 0 \), the parabola opens upward, potentially having a minimum point which depends on the function's vertices.
• Conversely, if \( a < 0 \), it opens downward, signifying a potential maximum point.
• Quadratic functions always have a single vertex, which can be found using vertex formula or derivative tests for finding minima or maxima.

In our exercise, the function given is cubic, but when finding critical points or slopes, we analyze its quadratic derivative \( f'(x) \). This derivative is used to determine local minima and adjust variables influencing the function's behavior.