Calculus help

Question

Indicate the domain over which the 𝑓(𝑥)=5𝑥^3−375𝑥 is decreasing.

Decreasing on

Answer 1

Take the derivative and find the intervals where the derivative is negative,

For f(x) = 5x³ - 375x, f '(x) = 15x² - 375

You can find out where the derivative equals zero and these would be the possible boundaries for where the function increases or decreases.

15x² - 375 =0

15(x² - 25) = 0

x² - 25 = 0

x² = 25

x = 5 and x = -5.

Put these values on a number line:

Notice how these divide the number line into 3 sections? You need to pick some number from each section and see what the sign of the slope is by plugging the numbers into the derivative function.

For the left interval (less than -5), let's pick x = -6. Plug in -6 into the derivative (f '(x) = 15x² - 375) so:

f '(-6) = 15(-6)² - 375) = 165 (which is a positive number) so the slope in this interval is positive

Now for the interval between -5 and 5, how about picking x = 0. Plug in 0 into the derivative so:

f '(0) = 15(0)² - 375) = -375 (which is a negative number) so the slope in this interval is negative.

For the right interval (greater than 5), let's pick x = 6. Plug in 6 into the derivative so:

f '(6) = 15(6)² - 375) = 165 (which is a positive number) so the slope in this interval is positive.

Intervals with positive slopes are increasing. Intervals with negative slopes are decreasing.

Mark your number line with corresponding plus or minus signs and associated increasing or decreasing notation like this:

Now you can use this to answer your question. Do not include the boundary as part of your interval (make it an open interval). The slope at the boundary point is zero so it is not increasing or decreasing at those points.