Find the largest interval on which the given function is increasing.


b. ƒ(x) = (x + 1)⁴

Verified step by step guidance

1. To determine where the function \( f(x) = (x + 1)^4 \) is increasing, we first need to find its derivative. The derivative, \( f'(x) \), will help us understand the behavior of the function.
2. Apply the power rule to differentiate \( f(x) = (x + 1)^4 \). The power rule states that \( \frac{d}{dx}[x^n] = nx^{n-1} \). Therefore, \( f'(x) = 4(x + 1)^3 \).
3. The function is increasing where its derivative is positive. So, we need to solve the inequality \( 4(x + 1)^3 > 0 \).
4. Since \( 4 \) is a positive constant, the inequality \( (x + 1)^3 > 0 \) determines where the function is increasing. A cubic expression is positive when \( x + 1 > 0 \), which simplifies to \( x > -1 \).
5. Thus, the largest interval on which the function \( f(x) = (x + 1)^4 \) is increasing is \( (-1, \infty) \).

Key Concepts

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is a fundamental tool in calculus used to determine the slope of the tangent line to the curve at any point. For a function to be increasing, its derivative must be positive over the interval in question.

Critical Points

Critical points occur where the derivative of a function is either zero or undefined. These points are essential for analyzing the behavior of the function, as they can indicate local maxima, minima, or points of inflection. To find intervals where the function is increasing, one must evaluate the derivative at these critical points.

Increasing and Decreasing Intervals

An interval is considered increasing if the function's output rises as the input increases, which occurs when the derivative is positive. Conversely, if the derivative is negative, the function is decreasing. By testing the sign of the derivative in the intervals defined by critical points, one can determine where the function is increasing.