SOLUTION: Perform a first derivative test on the function f(x)= x sqrt(100-x^2);[-10,10]. a. Locate the critical points of the given function. b. Use the first derivative test to locate th

Algebra ->  Test -> SOLUTION: Perform a first derivative test on the function f(x)= x sqrt(100-x^2);[-10,10]. a. Locate the critical points of the given function. b. Use the first derivative test to locate th      Log On


   

Question 1155432: Perform a first derivative test on the function f(x)= x sqrt(100-x^2);[-10,10].
a. Locate the critical points of the given function.
b. Use the first derivative test to locate the local maximum and minimum values.
c.identify the absolute minimum and maximum values of the function on the given intervals(when they exist)
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


f%28x%29+=+x%2Asqrt%28100-x%5E2%29+=+x%28100-x%5E2%29%5E%281%2F2%29

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.

Use the product rule to find the derivative.



df%2Fdx+=+%28100-2x%5E2%29%2F%28100-x%5E2%29%5E%281%2F2%29

The derivative is zero when the numerator is zero:

100-2x%5E2=0
100+=+2x%5E2
x%5E2+=+50
x+=+5sqrt%282%29 or x+=+-5sqrt%282%29

Find the function value at the critical point with the positive x value.

f%285sqrt%282%29%29+=+5sqrt%282%29%2Asqrt%28%28100-50%29%29+=+50

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).

graph%28400%2C400%2C-12%2C12%2C-80%2C80%2Cx%2Asqrt%28100-x%5E2%29%29