If $f (x) = (\frac{5}{2}) x^{\frac{2}{3}} - x^{\frac{5}{3}}$ $f(x) = (5/2)x^(2/3) - x^(5/3)$ , what are the points of inflection, concavity and critical points?

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Answer 1 · 2016-11-07T17:47:07

$f (x) = (\frac{5}{2}) x^{\frac{2}{3}} - x^{\frac{5}{3}}$ $f(x) = (5/2)x^(2/3) - x^(5/3)$

The domain of $f$ $f$ is $(- \infty, \infty)$ $(-oo,oo)$ .

$f'(x) = (5/3)x^(-1/3) - 5/3 x^(2/3)$

$= 5/3[x^(-1/3) - x^(2/3)]$

$= 5/3[(1-x)/x^(1/3)]$

$f'$ does not exist at $x=0$ and $f'(x) = 0$ at $x=1$ .
Both $0$ and $1$ are in the domain of $f$ , so both are critical numbers.

$f''(x) = 5/3[-1/3x^(-4/3)-2/3x^(-1/3)]$

$= -5/9x^(-4/3)[1+2x]$

$= -(5(1+2x))/(9x^(4/3))$

$f''(x) > 0$ for $x < -1/2$ and

$f''(x) > 0$ for $-1/2 < x < 0$ and $x > 0$

The graph of $f$ is concave up on $(-oo,-1/2)$ and concave down on $(-1/2,0)$ and on $(0,oo)$ .

The point $(-1/2,f(-1/2))$ is an inflection point.

If f(x) = (5/2)x^(2/3) - x^(5/3)f(x)=(52)x23−x53f(x) = (5/2)x^(2/3) - x^(5/3), what are the points of inflection, concavity and critical points?