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Draw the graph of y=x+(1)/(x)....

Draw the graph of `y=x+(1)/(x)`.

Text Solution

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We have `y=f(x) = |x|sqrt(1-x^(2))`
`f^(x)` is defined if `1-x^(2) ge0`
or `x in [-1,1]`
For `x ge0, f(x)=xsqrt(1-x^(2))`
`f(0)=0`
`f(x)=0 therefore x=0,1`
Thus, the graph meets the x-axis at `(0,0)` and `(1,0)`
Also `f(x) ge0` for `x ge0`
`f^(')(x) =1.sqrt(1-x^(2))-x (x)/sqrt(1-x^(2))`
`f^(')(x)=0 therefore x=1/sqrt(2)`, which is the point of maxima.
So the graph of the function for `x ge0` is as shown in the following figure.

When `x lt 0`, `y=-xsqrt(1-x^(2))`, graph of which can be obtained by reflecting the above graph in the y-axis.
Hence the graph of the function for its entire domain is as shown in the following figure.
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