Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = √(9 - x²) + sin⁻¹ (x/3)

Verified step by step guidance

1. First, find the derivative of the function f(x) = √(9 - x²) + sin⁻¹(x/3). Use the chain rule for the square root term and the derivative of the inverse sine function for the second term.
2. The derivative of √(9 - x²) is (1/2)(9 - x²)^(-1/2) * (-2x) = -x / √(9 - x²).
3. The derivative of sin⁻¹(x/3) is 1 / √(1 - (x/3)²) * (1/3) = 1 / (3√(1 - x²/9)).
4. Combine these derivatives to find f'(x): f'(x) = -x / √(9 - x²) + 1 / (3√(1 - x²/9)).
5. Determine where f'(x) is positive (indicating f is increasing) and where it is negative (indicating f is decreasing) by solving the inequality f'(x) > 0 and f'(x) < 0, respectively. Consider the domain restrictions from the square root and inverse sine functions.

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 for determining the slope of the tangent line to the curve at any point. By analyzing the sign of the derivative, we can identify intervals where the function is increasing (derivative > 0) or decreasing (derivative < 0).

Critical Points

Critical points occur where the derivative of a function is either zero or undefined. These points are essential for determining the behavior of the function, as they can indicate local maxima, minima, or points of inflection. To find intervals of increase or decrease, we evaluate the derivative at these critical points and test the sign of the derivative in the intervals they create.

Increasing and Decreasing Intervals

An increasing interval is a range of x-values where the function's output rises as x increases, while a decreasing interval is where the output falls. To find these intervals, we analyze the sign of the derivative across the critical points. If the derivative is positive in an interval, the function is increasing; if negative, it is decreasing.