F(x) = 3/4x^4 –x^3 -3x^2 + 6x

Determine the interval(s) where f(x) is increasing (if any) and the interval(s) where f(x) is decreasing (if any).

How do I know where f(x) is increasing or decreasing?

if f ' (x) > 0 , then the function is increasing

if f ' (x) < 0 , then the function is decreasing

so find the first derivative, and investigate it

Ohhhhhhhhh THANKS

To determine where a function is increasing or decreasing, you need to analyze the sign of the derivative of the function at different intervals.

To find the intervals of increase and decrease, we can take the derivative of the function f(x) with respect to x.

First, let's find the derivative of f(x):

f'(x) = d/dx (3/4x^4 – x^3 - 3x^2 + 6x)

To find the derivative, we apply the power rule and the sum/difference rule:

f'(x) = (3/4 * 4x^3) - (3x^2) - (6x^1) + 6 = 3x^3 - 3x^2 - 6x + 6

Now that we have the derivative, we need to determine the intervals where f'(x) > 0 (increasing) and f'(x) < 0 (decreasing). To do this, we find the critical points where f'(x) = 0 or is undefined.

To find critical points, we set f'(x) equal to zero and solve for x:

3x^3 - 3x^2 - 6x + 6 = 0

Unfortunately, solving this equation for x gives us a cubic equation, which is generally more complicated to solve. However, we can resort to graphical methods or use a calculator or software to solve for the critical points.

Let's use a graphing calculator or software to plot the function f(x) and determine where it is increasing or decreasing.

By examining the graph or using a software, we can identify the intervals where f(x) is increasing or decreasing. The intervals where f(x) is increasing will have positive slopes, while the intervals where f(x) is decreasing will have negative slopes.

Once you have found the critical points and plotted the graph, you can identify the intervals where f(x) is increasing or decreasing based on the sign of f'(x) within those intervals.