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How Many Solutions?

Find all the solutions to the this equation.

Quartics

Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.

Power Up

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

Slide

Age 16 to 18 Challenge Level:

Plot the graph of the function $y=f(x)$ where $f(x) = x(x+|x|)$. Find the first and second derivatives of the function. Show that the first derivative exists at $x=0$ but that the second derivative does not exist at $x = 0$.


NOTES AND BACKGROUND
This is a very simple question but it requires an understanding of how to handle functions that are defined differently on different parts of their domains.