# Analyse the function f(x) = (1+x)^(2/3)*(3-x)^(1/3) for extreme values, zeros and points of discontinuity, if any?

Mar 17, 2017

$f \left(x\right)$ has a local maximum of $\cong 2.117$ at $x = \frac{5}{3}$
$f \left(x\right)$ has zeros at $x = - 1$ and $x = 3$
$f \left(x\right)$ has a discontinuity at $\left(- 1 , 0\right)$

#### Explanation:

$f \left(x\right) = {\left(1 + x\right)}^{\frac{2}{3}} \cdot {\left(3 - x\right)}^{\frac{1}{3}}$

By inspection, $f \left(x\right)$ has zeros at $x = - 1$ and $x = 3$

Applying the product rule:

$f ' \left(x\right)$ = ${\left(1 + x\right)}^{\frac{2}{3}} \cdot \frac{1}{3} {\left(3 - x\right)}^{- \frac{2}{3}} \cdot \left(- 1\right) +$
$\frac{2}{3} {\left(1 + x\right)}^{- \frac{1}{3}} \cdot {\left(3 - x\right)}^{\frac{1}{3}}$

For points of extrema $f ' \left(x\right) = 0$

I.e. where: ${\left(1 + x\right)}^{\frac{2}{3}} \cdot \frac{1}{3} {\left(3 - x\right)}^{- \frac{2}{3}} \cdot \left(- 1\right) +$
$\frac{2}{3} {\left(1 + x\right)}^{- \frac{1}{3}} \cdot {\left(3 - x\right)}^{\frac{1}{3}} = 0$

$\frac{1}{3} {\left(1 = x\right)}^{\frac{2}{3}} \cdot {\left(3 - x\right)}^{- \frac{2}{3}} = \frac{2}{3} {\left(1 + x\right)}^{- \frac{1}{3}} \cdot {\left(3 - x\right)}^{\frac{1}{3}}$

${\left(1 + x\right)}^{\frac{2}{3}} / \left({\left(3 - x\right)}^{\frac{2}{3}}\right) = \frac{2 \cdot {\left(3 - x\right)}^{\frac{1}{3}}}{1 + x} ^ \left(\frac{1}{3}\right)$

Cross multiply:

${\left(1 + x\right)}^{\frac{2}{3} + \frac{1}{3}} = 2 \cdot {\left(3 - x\right)}^{\frac{1}{3} + \frac{2}{3}}$

$1 + x = 2 \left(3 - x\right) \to 1 + x = 6 - 2 x$

$3 x = 5$

$x = \frac{5}{3}$

Hence $f \left(x\right)$ has an extreme value at $x = \frac{5}{3}$

Now consider the graph of $f \left(x\right)$ below:

graph{(1+x)^(2/3)* (3-x)^(1/3) [-10, 10, -5, 5]}

We can see that $f \left(x\right)$ has a local maximum at $x = \frac{5}{3}$

$\therefore {f}_{\max} \left(x\right) = f \left(\frac{5}{3}\right) \cong 2.117$

Finally, considering the zeros of $f \left(x\right)$ we can see that $f \left(x\right)$ has a discontinuity at $\left(- 1 , 0\right)$