Calculus of a Single Variable, Chapter 3, 3.1, Section 3.1, Problem 45

Plot in MATLAB
>> f = 3.2*x^5 + 5*x^3 - 3.5*xf =(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.