1.3 Example

From [1, p. 128, 13g]

Asked: Graph

$$y = x \sqrt{x-1}$$

1.3.1 Solution

$$y = x \sqrt{x-1}$$

Factor $\sqrt{x-1}$ is $\sqrt{x}$ shifted one unit towards the right.

$$\epsffile{graphsx21.eps}$$

Multiplying by $x$ magnifies it by a factor ranging from 1 to $\infty$:

$$\epsffile{graphsx22.eps}$$

Function $y(x)$:

• has an $x$-extent $x$ $\raisebox{-.5pt}{$\geqslant$}$ 1 and a $y$-extent $y$ $\raisebox{-.5pt}{$\geqslant$}$ 0;
• behaves asymptotically as $y \sim x^{3/2}$ for $x\to \infty$;
• is monotonous:
  
  $$y' = \frac{{\rm d}y}{{\rm d}x} = \sqrt{x-1}+\frac{x}{2\sqr... ...rac{2x-2 + x}{2\sqrt{x-1}} = \frac{3x-2}{2\sqrt{x-1}} > 0;$$
  
  

• has vertical slope at $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1;
• is concave down for smaller $x$, concave up for larger $x$;
• the inflection point is at
  
  $$y'' = \frac{3x-4}{4(x-1)^{3/2}}=0$$
  
  
  giving $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $4/3$.