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My second derivative is

$$\frac{(250-64 x^3)}{(3 x (-125+8 x^3))^\frac{5}{3}} $$ I know that function is undefined if $x = 0$ and $x=5/2$. What would be concavity intervals? Why not $(-\infty,0)$,$(0,5/2)$ - Increasing and $(5/2,\infty)$ - decreasing ?

bryan.blackbee
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Sergey Tsibel
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  • The numerator also affects the sign. The places where there could be a change in sign, and therefore of concavity, include the cube root of $250/64$. – André Nicolas Mar 06 '13 at 07:58
  • ah, yes yes. Even with the extra point 5/(2 2^(2/3)), I'm still not able to solve it – Sergey Tsibel Mar 06 '13 at 08:02
  • Increasing (-INF,0)(0,root) Decreasing (root,5/2),(5/2,INF) Something wrong? My first derivative was (8x^4-125x)^(-2/3) – Sergey Tsibel Mar 06 '13 at 08:06
  • What was the original function? – André Nicolas Mar 06 '13 at 08:11
  • I didn't have one. Just a derivative of the function. So, my increasing interval (-inf,0) doesn't seems to work. As well as my decreasing (0,root),(root,5/2),(5/2,inf) – Sergey Tsibel Mar 06 '13 at 08:15
  • Here is correct calculation. For $x\lt 0$, top is negative, bottom positive, concave down. For $0\lt x\lt (125/32)^{1/3}$, top negative, bottom negative, concave up. then up to $5/2$ top positive, bottom negative, concave down. After $5/2$, everything positive, concave up. (I had wrong sign before.) – André Nicolas Mar 06 '13 at 08:24
  • You are absolutely right. Shame on me for not being able to solve this problem myself. – Sergey Tsibel Mar 06 '13 at 08:32
  • Minus signs are a headache. I have trouble with them. – André Nicolas Mar 06 '13 at 08:34

1 Answers1

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The top is negative up to $x=(125/32)^{1/3}\approx 1.575$. The bottom is positive for $x\lt 0$, negative for $0\lt x\lt 5/2$, and positive for $x\gt 5/2$.

Thus for $x\lt 0$, the second derivative is negative, and therefore the first derivative is decreasing, the graph is concave down.

From $0$ to $(125/32)^{1/3}$, the top is still negative. The bottom is negative, so the ratio is positive, the graph of our function is concave up (some people say convex).

From $(125)^{1/3}$, for similar reasons the graph is concave down.

Finally, past $5/2$, the curve is concave up.

André Nicolas
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