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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:

Why do this problem?
It requires an understanding of how to handle functions that are defined differently on different parts of their domains and how to interpret the situation when the derivative takes different values close to point but on opposite sides of the point.

Possible approach
A short problem that can be used as a lesson starter.

Key questions
What can you say about the function when $x< 0$?

What can you say about the function when $x> 0$?