Question

Given the function f(x)=x²/³ (6−x)¹/³

Find the following: (a) the intervals where f(x) is increasing and where f(x) is decreasing:

Answer 1

To find out where the function is increasing or decreasing, we differentiate it, set the first derivative equal to zero to find critical points, and examine the sign of the derivative over different intervals.

Step-by-step explanation:

To determine where the function f(x) = x^{\frac{2}{3}}(6 - x)^{\frac{1}{3}} is increasing or decreasing, we need to find its first derivative and analyze the sign of the derivative.

Step 1: Differentiate f(x)

We use the product rule for differentiation, which states that for two functions u(x) and v(x), the derivative of their product u(x)v(x) is given by u'(x)v(x) + u(x)v'(x).

Step 2: Set the derivative equal to zero

We find the critical points of f(x) by setting the derivative equal to zero and solving for x. These points are where f(x) may change from increasing to decreasing or vice versa.

Step 3: Determine sign of the derivative

We create a number line with the critical points and test the sign of the derivative in each interval. If the derivative is positive, f(x) is increasing; if it's negative, f(x) is decreasing.