Let f(x)= x^5/120 - x^3/6 + x
Find the local minima and maxima, then determine where the function is decreasing and increasing.
Basically, I just need to know how to type this correctly in my calculator so I can get the right minima and maxima.
I had:
local minima: f(-1.59)=-1.00
f(3.08)=.520
local maxima: f(1.59)=1.00
f(-3.08)=-.520
and
increasing : (-1.5, 1.5)
decreasing : (1.5, 3)
Hmm I do not know how to do it entirely by calculator since you have to find the derivative by hand. You can then plug the derivative into the calculator and get the zeros for the local minima and maxima. Local minimas in this case occur only when the slope changes from negative to positive. Local maximas occur when the slope changes from positive to negative. BTW, I took a quick look at the graph and you are missing some intervals for increasing and decreasing so you may want to double check them.
To find the local minima and maxima of the function f(x) = (x^5/120) - (x^3/6) + x, you can follow these steps:
Step 1: Calculate the derivative of the function f(x) with respect to x. The derivative will give you information about the slope of the function at different points.
Taking the derivative of f(x) gives: f'(x) = (5x^4/120) - (3x^2/6) + 1
Simplifying further, f'(x) = (x^4/24) - (x^2/2) + 1
Step 2: Set the derivative f'(x) equal to zero to find the critical points where the slope of the function changes. Solve the equation f'(x) = 0 for x.
(x^4/24) - (x^2/2) + 1 = 0
This equation can be quite complex to solve manually, so you can use your calculator to find the approximate solutions.
Step 3: Enter the equation f'(x) = (x^4/24) - (x^2/2) + 1 into your calculator's equation solver or function graphing feature. Make sure you set the equation equal to zero.
For the interval you are interested in regarding local minima and maxima, it seems to be around -3 to 3. So set your calculator to solve the equation in that interval.
Step 4: Use your calculator to find the approximate solutions to the equation f'(x) = 0. These solutions correspond to the x-values of the critical points.
Based on your calculations, the solutions you found were:
x ≈ -3.08
x ≈ -1.59
x ≈ 1.59
x ≈ 3.08
Step 5: Plug these critical points (x-values) back into the original function f(x) to find the corresponding y-values.
Evaluate f(-3.08), f(-1.59), f(1.59), and f(3.08) using your calculator.
Based on your calculations, you obtained the following local minima and maxima:
Local minima:
f(-1.59) ≈ -1.00
f(3.08) ≈ 0.520
Local maxima:
f(1.59) ≈ 1.00
f(-3.08) ≈ -0.520
Now, to determine where the function is increasing and decreasing, you can analyze the intervals between the critical points.
Based on your calculations, it seems that the function is decreasing for the interval (-1.5, 1.5) and increasing for the interval (1.5, 3).
Hence, the final answer to your question is:
Local minima: (approximately) (-1.59, -1.00) and (3.08, 0.520)
Local maxima: (approximately) (1.59, 1.00) and (-3.08, -0.520)
Increasing interval: (1.5, 3)
Decreasing interval: (-1.5, 1.5)
By following these steps and using your calculator's equation solver, you should be able to obtain the correct minima, maxima, and intervals for any function.