Sanjeet sent us this graph, showing the
three curves on the same graph:
For different values of a the graphs have the equation: y =
{a^3\over (x^2+a^2)} Differentiating {dy\over dx} = -
{2a^3x\over (x^2+a^2)^2}. So the critical value is at (0,a).
Near this point, for small negative x the gradient is positive
and the function increasing and for small positive x the gradient
is negative and the function decreasing so this is a maximum
point.
The value of y is always positive so the entire graph lies above
the x axis and the graph is symmetrical about x=0 because
f(-x)=f(x).
For large x the value of y tends to zero so the line y=0 is
an asymptote.
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