Question 1155432
<br>
{{{f(x) = x*sqrt(100-x^2) = x(100-x^2)^(1/2)}}}<br>
We can note before we start that the function is odd -- that is, f(-x) = -f(x).  We can use that, if we find it convenient, to simplify solving the problem.<br>
Use the product rule to find the derivative.<br>
{{{df/dx = ((x)(1/2)(-2x))/(100-x^2)^(1/2)+(100-x^2)^(1/2)}}}
{{{df/dx = (-x^2)/(100-x^2)^(1/2)+(100-x^2)/(100-x^2)^(1/2)}}}
{{{df/dx = (100-2x^2)/(100-x^2)^(1/2)}}}<br>
The derivative is zero when the numerator is zero:<br>
{{{100-2x^2=0}}}
{{{100 = 2x^2}}}
{{{x^2 = 50}}}
{{{x = 5sqrt(2)}}} or {{{x = -5sqrt(2)}}}<br>
Find the function value at the critical point with the positive x value.<br>
{{{f(5sqrt(2)) = 5sqrt(2)*sqrt((100-50)) = 50}}}<br>
The maximum value of the function is at (5*sqrt(2),50); since the function is an odd function, the minimum value is at -5*sqrt(2),-50).<br>
{{{graph(400,400,-12,12,-80,80,x*sqrt(100-x^2))}}}<br>