`f(x) = 3.2x^5 + 5x^3 - 3.5x, [0,1]` (a) Use a computer algebra system to graph the function and approximate any absolute extrema on the given interval. (b) Use the utility to find any...

`f(x) = 3.2x^5 + 5x^3 - 3.5x, [0,1]` (a) Use a computer algebra system to graph the function and approximate any absolute extrema on the given interval. (b) Use the utility to find any critical numbers, and use them to find any absolute extrema not located at the endpoints. Compare the results with those in part (a).

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Textbook Question

Chapter 3, 3.1 - Problem 45 - Calculus of a Single Variable (10th Edition, Ron Larson).
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thilina-g | College Teacher | (Level 1) Educator

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Plot in MATLAB

>> f = 3.2*x^5 + 5*x^3 - 3.5*x
 
f =
 
(16*x^5)/5 + 5*x^3 - (7*x)/2

a) See the attached graph

There seems to be a local minima between 0.4 and 0.5

b)

Roots of f

x = 0, x = -0.7240, x = 0.7240, x = 0.0000 - 1.4445i and x = 0.0000 + 1.4445i. Two of them are imagianry roots.

Asymptotes-

>> limit(f, -inf)
 
ans =
 
-Inf
 
>> limit(f, inf)
 
ans =
 
Inf

There are no horizontal asymptotes.

Maxima and minima-

Find the derivative and find roots of it.

>> f1=diff(f)
 
f1 =
 
16*x^4 + 15*x^2 - 7/2

There are two real roots for this equation, at 0.4398 and -0.4398. 0.4398 is the local minima we saw in the graph.

We want to know the inflections points now.

Let's find the derivative of f1 and find roots of it.

>> f2= diff(f1)
 
f2 =
 
64*x^3 + 30*x
 

There is a real root at 0. This is the inflection point we are looking for.

The results show that there are two other critical points beyond the range of [0,1], a minima at -0.4398 and an inflection point at 0.

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