Calculus Help

[deathsta2]

Hey,
I have quiz this Friday on using derivatives to analyze and develop graphs of functions by developing sign charts to assist in the analysis of functions.

Question: f(x)= cube root of (x^2+x+1)

Here's how far I got...

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

f''(x)=2/3(x^2+x+1)^-1/3 + 2/3x(-1/3(x^2+x+1)^-4/3)
= ?
Can I simplify to 2/3(x^2+x+1)^-1/3 * (x^2+x+1) by factoring out 2/3(x^2+x+1)^-1/3

Then after applying the sign charts... I get wrong answers for (a) the maximum/minimum values, (b)the intervals on which f increasing,(c) the intervals on which f decreasing, (d) open intervals on which f is concave up,(e) open intervals on which f is concave down, (f) the x-coordinates of all inflection points.

I think I miscalculated the second derivative.

[raulperez]

not to sound like an *sshole or anything, but honestly you miscalculated just about everything

in the first derivative you forgot chain rule and mixed up the resultant power and the coefficient (they should be swapped, although the negative stays on the power)
chain rule: if h(x)=f(g(x)),then h'(x)=f'(g(x))g'(x)
d(x^(1/3)) = (1/3)(x^(-2/3))
thus:

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

now for this you have to use product rule: if h(x)=f(x)g(x), then h'(x)=f(x)g'(x)+f'(x)g(x)
therefore:

f"(x)=(-2/9)((x^2+x+1)^(-5/3))*((2x+1)^2)+(2/3)(x^2+x+1)^(-2/3)

[deathsta2]

Hey thanks. I think I was f*cking drunk or something when I posted that.

Also... possible *sshole:

Spoiler: show

raulperez

[raulperez]

lol i just tell it how it is, how'd you do with the quiz?

[deathsta2]

Not sure yet, I get it back on Monday. I think I did pretty well... surely no less than 90%.