Expert: Paul Klarreich - 2/14/2009

Question
4.) Consider the function f(x)= x^(4/3) + 4x^(1/3) on the interval -8 ≤ x ≤ 8.
a.) Find the coordinates of all points at which the tangent to the curve is a horizontal line.
b.) Find the coordinates of all points at which the tangent to the curve is a vertical line.
c.) Find the coordinates of the points at which the absolute maximum and minimum occur.
d.) For what values of x is the function concave down?

Answer
Questioner:   Linrosa
Category:  Calculus
Private:  No
 
Subject:  Tangents, Max & Min
Question:  4.) Consider the function f(x)= x^(4/3) + 4x^(1/3) on the interval -8 ≤ x ≤ 8.
a.) Find the coordinates of all points at which the tangent to the curve is a horizontal line.
b.) Find the coordinates of all points at which the tangent to the curve is a vertical line.
c.) Find the coordinates of the points at which the absolute maximum and minimum occur.
d.) For what values of x is the function concave down?

You start with  

f'(x) = 4(x^1/3)/3 + 4x^(-2/3)/3

f'(x) = 4/3[ x^1/3 + 4x^(-2/3)]  << use this later.

f'(x) = 4/3[ x^1/3 + 1/x^(2/3)]  

f'(x) = 4/3[ x^1/3 + 1/x^(2/3)]

f'(x) = 4/3[ (x + 1)/x^(2/3)]

Now you can do your stuff:


a.) Find the coordinates of all points at which the tangent to the curve is a horizontal line.

Set f'(x) = 0.  Solve, then substitute into f(x) to get y.

b.) Find the coordinates of all points at which the tangent to the curve is a vertical line.

Find where f'(x) is undefined.  Substitute, if possible.

c) Use the results of a and b, along with your endpoints -8, 8.  Test f(each of them)

d) Obtain f''(x) from "use this later".  Find the intervals between where:

f'' = 0
f'' is undefined
endpoints.

Concave down means f'' < 0.

[In a couple of days I'll look at the other one.]

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