How's this for a test question?

[hank]

For the function:
f(x) = 2x^(5/3) - 5x^(2/3)

Find all positive and negative intervals, positive and negative concavity, and all points of differentiation.

As if the problem wasn't tedious enough, let's throw in some fractional exponents.

Is it just me, or was this problem a little excessive?
Geeze.

I'm sure I made mistakes on the 2d derivative with the fractions. Very annoying.

Anyway, here's another question.
Find all points on the interval [0, 2pi] where the tangent line of x - 2cosx is horizontal.
This one I actually wouldn't mind help with.
Here's what I did...

//Find the derivative
f'(x) = 1 + 2sinx

//Set that to 0 and solve for x
1 + 2sinx = 0
1 = -2sinx
1/2 = -sinx
x = 5pi/6

//Plug x into original equation to get y
y = 5pi/6 - 2cos(5pi/6)
y = 5pi/6 + 2(sqrt(3)/2)
y = 5pi/6 + sqrt(3)
y = 5pi/6 + 6sqrt(3)/6
y = (5pi + 6sqrt(3)) / 6

So coords for a horizontal tangent line: (5pi/6, (5pi + 6sqrt(3)) / 6)

I'm almost certain this is wrong, and I believe my mistake is when I differentiated and had to find the angle for x.

Can someone double-check my work?

I so bombed this test, sigh.

Just noticed, x should probably be pi/6 and 5pi/6.

 

[hank]

Actually, think I figured it out.
For the derivative set to 0:

1 + 2sinx = 0
2sinx = -1
sinx = -1/2
x = 7pi/6 and 11pi/6

//Plug in x and solve for y:
7pi/6 -2cos(7pi/6) = 7pi/6 + sqrt(3)

and

11pi/6 - 2cos(11pi/6) = 11pi/6 - sqrt(3)

//Therefore on the interval [0, 2pi], points where tangent line is horizontal:
(7pi/6, 7pi/6 - sqrt(3)) and (11pi/6, 11pi/6 - sqrt(3))

How about that?

 

[tkhunny]

For the function:
f(x) = 2x^(5/3) - 5x^(2/3)

Find all positive and negative intervals, positive and negative concavity, and all points of differentiation.

This is a fine problem, challenging your understanding of many concepts.

What did you get for the first two derivatives?

The only REAL problem I see is trying to graph it with a calculator or computer program. These may miss all the stuff where x < 0. You would not want to miss that half the graph.

 

[hank]

f'(x) = 1 + 2sinx

and

f''(x) = 2cosx

 

[hank]

bump