Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.


ƒ(x) = eˣ + e⁻ˣ

Verified step by step guidance

1. To find the critical points of the function \( f(x) = e^x + e^{-x} \), we first need to find its derivative. The derivative of \( e^x \) is \( e^x \), and the derivative of \( e^{-x} \) is \( -e^{-x} \). Therefore, the derivative \( f'(x) \) is \( e^x - e^{-x} \).
2. Set the derivative equal to zero to find the critical points: \( e^x - e^{-x} = 0 \).
3. To solve \( e^x - e^{-x} = 0 \), add \( e^{-x} \) to both sides to get \( e^x = e^{-x} \).
4. Recognize that \( e^x = e^{-x} \) implies \( e^{2x} = 1 \) by multiplying both sides by \( e^x \).
5. Solve \( e^{2x} = 1 \) by taking the natural logarithm of both sides, which gives \( 2x = 0 \). Therefore, \( x = 0 \) is the critical point.

Key Concepts

Critical Points

Critical points of a function occur where its derivative is either zero or undefined. These points are essential for identifying local maxima, minima, and points of inflection. To find critical points, one must first compute the derivative of the function and then solve for the values of x that satisfy the condition of the derivative being zero.

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus that provides information about the function's slope at any given point. For the function ƒ(x) = eˣ + e⁻ˣ, the derivative can be calculated using the rules of differentiation, particularly the exponential function's properties.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a constant and 'x' is the variable. They are characterized by their rapid growth or decay and are crucial in various applications, including modeling population growth and radioactive decay. In the given function, eˣ and e⁻ˣ are examples of exponential functions, which will influence the behavior of the derivative and the location of critical points.