Problem 34 - A Textbook of Differential Calculus

Question

Domain of the function \(f(x)\), if \(3^{x}+3^{f(x)}=\) minimum of \(\phi(t)\) where \(\phi(t)=\) minimum of \(\left\\{2 t^{3}-15 t^{2}+36 t-25,2+|\sin t| ; 2 \leq t \leq 4\right\\}\) is (a) \((-\infty, 1)\) (b) \(\left(-\infty, \log _{3} e\right)\) (c) \(\left(0, \log _{3} 2\right)\) (d) \(\left(-\infty, \log _{3} 2\right)\)

Step-by-Step Solution

Answer

The domain of the function is \((-∞, \log_3 2)\).

Step 1: Define the Minimum Function

Let's first analyze what the minimum of \( \phi(t) \) is. The function \( \phi(t) = \min\{ 2t^3 - 15t^2 + 36t - 25, 2 + |\sin t| \} \) needs to be evaluated over the interval \( 2 \leq t \leq 4 \). To find where these functions intersect or achieve their minimum, we need to check these two expressions separately over the interval.

Step 2: Evaluate the Polynomial Function

The polynomial \( 2t^3 - 15t^2 + 36t - 25 \) must be evaluated over the interval \( [2, 4] \). Calculate the value of this polynomial at critical points and endpoints within this range to find where it reaches its minimum.

Step 3: Evaluate the Trigonometric Function

The function \( 2 + |\sin t| \) must also be evaluated over the interval \( [2, 4] \). Calculate the minimum value within this range by looking for the minimum value of \( |\sin t| \) in this interval and how it impacts \( 2 + |\sin t| \).

Step 4: Find the Overall Minimum

Compare the minimum values found from \( 2t^3 - 15t^2 + 36t - 25 \) and \( 2 + |\sin t| \) over the interval \( [2, 4] \). The function \( \phi(t) \) will take the smaller value from these two computations.

Step 5: Solve the Equation for Domain

Given \( 3^x + 3^{f(x)} = \min(\phi(t)) \), set \( f(x) = c \) such that \( 3^x + 3^c = m \), where \( m = \) minimum of \( \phi(t) \). Solve for \( x \) to determine its range taking into account \( m \) and simplifying accordingly. If \( m \) allows \( 3^c \) be a positive, calculate \( 3^x < m \). Simplifying, we set the domain of \( x \).

Step 6: Express the Domain

The intersection of the possible solutions for \( x \), consistent with both the functions and the constraint \( m \), provides the domain of \( f(x) \). Given that the smallest value occurs is \( 2 + |\sin 4| \), which is \( 3 \), we set \( 3^x < 2 \), giving the domain \((-∞, \log_3 2)\).

Key Concepts

Understanding Minimum Functions

In mathematics, the concept of the minimum function is vital for solving problems involving optimization. It allows us to find the smallest value attained by a function over a certain interval or domain. For the function \( \phi(t) = \min\{2t^3 - 15t^2 + 36t - 25, 2 + |\sin t|\} \), we need to determine the smallest output within the interval \([2, 4]\).
To find the minimum, you must evaluate both expressions separately:

• The polynomial \( 2t^3 - 15t^2 + 36t - 25 \)
• The trigonometric expression \( 2 + |\sin t| \)

After computing these, the function \( \phi(t) \) will take the smallest value from these computations over the given interval. It's important to approach this by calculating critical points and endpoints to ascertain precisely where the minimal value lies.

Trigonometric Function Evaluation

Evaluating a trigonometric function like \(|\sin t|\) over an interval requires understanding its periodic behavior. Since \( \sin t \) oscillates between -1 and 1, \(|\sin t|\) takes values between 0 and 1.
Within the specified interval \([2, 4]\), we evaluate the expression \(2 + |\sin t|\). This function changes as \(t\) changes; thus, you should check specific values of \(t\) to determine the effect on the minimum value across the interval.
Steps to evaluate include:

• Identify maximum and minimum values of \(|\sin t|\) between \(t = 2\) and \(t = 4\).
• Calculate \(2 + |\sin t|\) at these critical points.

This process helps determine how low the expression \(2 + |\sin t|\) can go in the interval, helping to find the minimum.

Polynomial Function Analysis

Analyzing a polynomial function involves identifying its critical points and understanding its behavior over an interval. The polynomial \(2t^3 - 15t^2 + 36t - 25\) requires solving for

• Critical points: Where the derivative is zero or undefined.
• Endpoints: The function values at the start and end of the interval \([2, 4]\).

To find the minimum of this polynomial within the interval, calculate derivatives using basic calculus to find critical points and substitute these back into the polynomial to see which value is smallest. Evaluate it also at the interval's endpoints \(t = 2\) and \(t = 4\) to ensure all possible minimum positions are considered.

Solving Inequalities

Once the minimum value of a function is identified, solving inequalities becomes a method to determine the domain of another function. In this exercise, we're solving for when \(3^x + 3^{f(x)} = m\), with \(m\) being the defined minimum.
The goal is:

• Set \(3^x < m\: 3^c\).
• Solve for \(x\) to determine its range.

This involves using logarithmic functions to isolate \(x\) and express it in terms of \(m\). The logarithm base will be 3 due to the powers involved \(3^x\) making our inequalities and calculations centered on finding \(\log_3\) values. Finally, the domain for \(x\) consistent with all constraints given by \(m\) remains as the solution, noting any endpoints or critical values from previous steps in our computations.