when f(x)=2x^3+13x^2−48x−27, Find the relative maximum and minimum, then the interval notation of where the function increases and decreases

Answer 1

Since you have this listed under the categories of "Math" "Algebra 2" and "trigonometry", I'm going to assume you do NOT wish to use Calculus skills to solve this problem.

The best way to do this problem is to use your skills with your graphing calculator. I'm going to assume you have a TI-84. If not, you can use desmos.

Graph the function by typing it into your "Y₁" by pushing the "y=" button and typing in:

Y₁ = 2x³ + 13x² − 48x − 27 and then pushing the "graph" button. The graph will most likely by mostly off your screen so you'll need to set the window properly in order to see it. Push the "window" button. I used to following settings:

Xmin= -12

Xmax= 8

Xscl=1

Ymin= -100

Ymax= 400

Yscl= 1

Xres= 1

ΔX= 0.1

TraceStep= 0.2

My graph looks like this:

To find the maximum, push the blue "2nd" button, then the "trace" button and scroll down to "4: maximum" and select it. The calculator then asks you the question "Left Bound?". Using your cursor buttons scroll on the curve until your cursor is just left of the maximum (around x = -7) and select "enter'. Then your calculator asks you the question "Right Bound?" and you scroll your cursor just to the right of the maximum (around x = -4) and select "enter". Then your calculator asks you the question "Guess?" and you scroll your cursor left until you are approximately on the maximum (around x = -5.8) and select "enter". The calculator then gives you the maximum as: X = -5.729593 Y=298.60267

Repeat this process on the minimum (of course, selecting "3:minimum" from the menu.

The intervals where the function is increasing and decreasing are easy once you have these points. The function is increasing on the interval (-∞, -5.729593) is an example of what you will get. Note that these are the x-values where the function increases. You will find another interval where the function is decreasing and yet another interval where the function is increasing.