For What Values Of $a$ Do The Intervals On Which The Functions ${ F(x) = \frac{4x - 6}{e^x} }$and ${ G(x) = -x^2 + Ax + 2 }$are Increasing Coincide?

For What Values of $a$ Do the Intervals on Which the Functions $f(x)$ and $g(x)$ Are Increasing Coincide?

In this article, we will explore the values of $a$ for which the intervals on which the functions $f(x) = \frac{4x - 6}{e^x}$ and $g(x) = -x^2 + ax + 2$ are increasing coincide. This involves finding the intervals where both functions are increasing and determining the values of $a$ that make these intervals identical.

Increasing Intervals of $f(x)$

To find the increasing intervals of $f(x)$, we need to find the critical points of the function. The critical points occur when the derivative of the function is equal to zero or undefined.

Derivative of $f(x)$

The derivative of $f(x)$ is given by:

$$f'(x) = \frac{(4x - 6)e^x - (4)e^x}{(e^x)^2}$$

Simplifying the derivative, we get:

$$f'(x) = \frac{4xe^x - 6e^x - 4e^x}{e^{2x}}$$

$$f'(x) = \frac{4xe^x - 10e^x}{e^{2x}}$$

Critical Points of $f(x)$

To find the critical points, we set the derivative equal to zero:

$$\frac{4xe^x - 10e^x}{e^{2x}} = 0$$

Since $e^{2x}$ is always positive, we can divide both sides by $e^{2x}$:

$$4xe^x - 10e^x = 0$$

Factoring out $e^x$, we get:

$$e^x(4x - 10) = 0$$

This gives us two possible critical points:

$$e^x = 0 \quad \text{or} \quad 4x - 10 = 0$$

Since $e^x$ is never equal to zero, we only consider the second critical point:

$$4x - 10 = 0$$

Solving for $x$, we get:

$$x = \frac{10}{4} = \frac{5}{2}$$

Increasing Intervals of $f(x)$

To determine the increasing intervals of $f(x)$, we need to test the intervals on either side of the critical point.

Let's test the interval $(-\infty, \frac{5}{2})$:

• For $x < \frac{5}{2}$, we have $4x - 10 < 0$, so $e^x(4x - 10) < 0$.
• For $x > \frac{5}{2}$, we have $4x - 10 > 0$, so $e^x(4x - 10) > 0$.

Since the derivative changes sign at $x = \frac{5}{2}$, we know that $f(x)$ is increasing on the interval $(\frac{5}{2}, \infty)$.

Increasing Intervals of $g(x)$

To find the increasing intervals of $g(x)$, we need to find the critical points of the function. The critical points occur when the derivative of the function is equal to zero or undefined.

Derivative of $g(x)$

The derivative of $g(x)$ is given by:

$$g'(x) = -2x + a$$

Critical Points of $g(x)$

To find the critical points, we set the derivative equal to zero:

$$-2x + a = 0$$

Solving for $x$, we get:

$$x = \frac{a}{2}$$

Increasing Intervals of $g(x)$

To determine the increasing intervals of $g(x)$, we need to test the intervals on either side of the critical point.

Let's test the interval $(-\infty, \frac{a}{2})$:

• For $x < \frac{a}{2}$, we have $-2x + a > 0$, so $g(x)$ is increasing.
• For $x > \frac{a}{2}$, we have $-2x + a < 0$, so $g(x)$ is decreasing.

Since the derivative changes sign at $x = \frac{a}{2}$, we know that $g(x)$ is increasing on the interval $(-\infty, \frac{a}{2})$.

Coinciding Increasing Intervals

To find the values of $a$ for which the increasing intervals of $f(x)$ and $g(x)$ coincide, we need to find the values of $a$ that make the critical points of both functions identical.

From the previous sections, we know that the critical point of $f(x)$ is $x = \frac{5}{2}$, and the critical point of $g(x)$ is $x = \frac{a}{2}$.

Setting the critical points equal to each other, we get:

$$\frac{5}{2} = \frac{a}{2}$$

Solving for $a$, we get:

$$a = 5$$

Therefore, the values of $a$ for which the increasing intervals of $f(x)$ and $g(x)$ coincide are $a = 5$.

In this article, we explored the values of $a$ for which the intervals on which the functions $f(x) = \frac{4x - 6}{e^x}$ and $g(x) = -x^2 + ax + 2$ are increasing coincide. We found that the increasing intervals of $f(x)$ are $(\frac{5}{2}, \infty)$, and the increasing intervals of $g(x)$ are $(-\infty, \frac{a}{2})$. We then found that the values of $a$ for which the increasing intervals of $f(x)$ and $g(x)$ coincide are $a = 5$.

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart

Derivative of $f(x)$

The derivative of $f(x)$ is given by:

$$f'(x) = \frac{(4x - 6)e^x - (4)e^x}{(e^x)^2}$$

Simplifying the derivative, we get:

$$f'(x) = \frac{4xe^x - 6e^x - 4e^x}{e^{2x}}$$

$$f'(x) = \frac{4xe^x - 10e^x}{e^{2x}}$$

Critical Points of $g(x)$

To find the critical points, we set the derivative equal to zero:

$$-2x + a = 0$$

Solving for $x$, we get:

$$x = \frac{a}{2}$$

Increasing Intervals of $g(x)$

To determine the increasing intervals of $g(x)$, we need to test the intervals on either side of the critical point.

Let's test the interval $(-\infty, \frac{a}{2})$:

• For $x < \frac{a}{2}$, we have $-2x + a > 0$, so $g(x)$ is increasing.
• For $x > \frac{a}{2}$, we have $-2x + a < 0$, so $g(x)$ is decreasing.

Since the derivative changes sign at $x = \frac{a}{2}$, we know that $g(x)$ is increasing on the interval $(-\infty, \frac{a}{2})$.
Q&A: For What Values of $a$ Do the Intervals on Which the Functions $f(x)$ and $g(x)$ Are Increasing Coincide?

A: The main goal of this article is to find the values of $a$ for which the intervals on which the functions $f(x) = \frac{4x - 6}{e^x}$ and $g(x) = -x^2 + ax + 2$ are increasing coincide.

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A: The increasing intervals of $f(x)$ are $(\frac{5}{2}, \infty)$.

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A: The increasing intervals of $g(x)$ are $(-\infty, \frac{a}{2})$.

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A: We find the values of $a$ by setting the critical points of both functions equal to each other. In this case, we set $x = \frac{5}{2} = \frac{a}{2}$ and solve for $a$.

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A: The value of $a$ for which the increasing intervals of $f(x)$ and $g(x)$ coincide is $a = 5$.

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A: It is important to find the values of $a$ because it allows us to determine the intervals on which both functions are increasing. This is useful in a variety of applications, such as optimization problems and data analysis.

A: Some real-world applications of finding the increasing intervals of functions include:

• Optimization problems: Finding the maximum or minimum value of a function is a common problem in optimization.
• Data analysis: Understanding the behavior of a function can help us make predictions and identify trends in data.
• Physics and engineering: Many physical systems can be modeled using functions, and understanding the behavior of these functions is crucial in designing and optimizing these systems.

A: We can use the results of this article to:

• Optimize functions: By understanding the increasing intervals of a function, we can identify the optimal values of the function.
• Analyze data: By understanding the behavior of a function, we can make predictions and identify trends in data.
• Design and optimize physical systems: By understanding the behavior of functions that model physical systems, we can design and optimize these systems.

A: Some potential limitations of this article include:

• The article assumes that the functions $f(x)$ and $g(x)$ are continuous and differentiable.
• The article assumes that the increasing intervals of $f(x)$ and $g(x)$ are identical.
• The article does not consider other factors that may affect the behavior of the functions, such as external influences or non-linear effects.

A: Some potential future directions for this research include:

• Investigating the behavior of functions with multiple variables.
• Considering the effects of external influences on the behavior of functions.
• Developing new methods for analyzing and optimizing functions.