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How Many Solutions?

Find all the solutions to the this equation.


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

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Age 16 to 18 Challenge Level:

Drawing the graphs first will greatly aid in their interpretation!

Key to this problem is the idea for small values of $t$ we have $t$ is a lot larger than $t^2$, which is in turn a lot larger than $t^3$. For large values of $t$ the reverse is true.