Posted by **Joshua** on Monday, April 20, 2009 at 12:58pm.

I'm having a lot of trouble with this problem:

Sketch the graph and show all local extrema and inflections.

f(x)= (x^(1/3)) ((x^2)-175)

I graphed the function on my graphing calculator and found the shape.

I also found the first derivative:

(7/3)(x^(4/3)) - (175/3)(x(-2/3))

But the number I found for x (when I set the deriv. to 0 to find the critical points) was 1.71, but this doesn't correspond to a min or a max. Then I plugged the equation of the first derivative in onto my calculator, and the corresponding y value of 1.71 was NOT zero.... furthermore, it seemed like there was a horizontal asymptote at 0, because the derivative never was 0.

I couldn't find an inflection point, because the second derivative never equaled zero.

Does this seem right-- that there is no local max, min, OR inflection point? If this is wrong, can you help me find the right values? I sketched the graph on my paper but I still haven't found the min, max, or inflection points.

- calc. -
**Reiny**, Monday, April 20, 2009 at 2:12pm
I had the same first derivative as you did

y' = (7/3)(x^(4/3)) - (175/3)(x(-2/3))

= (1/3)x^(-2/3)[7x^2 - 175] by factoring

setting this equal to zero, ...

the first factor of (1/3)x^(-2/3) yields no solution but

7x^2-175=0 gives me

x = ± 5

my second derivative was (28/9)x^(1/3) + (350/9)x^(-5/3) which when set to zero has no answer.

so according to the math, there should be 2 points of max/min, but no point of inflection.

Does that fit in with your sketch?

I did not sketch the graph.

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