Question

Consider the following.
f(x) = 8x(square root of (x − x^2)) Use a graph to find the absolute maximum and minimum values of the function to two decimal places.

Answer 1

only valid for 90 ≤ x ≤ 1

http://www.wolframalpha.com/input/?i=y+%3D+%3D+8x%28√%28x+−+x%5E2%29%29+%2C+0+%3C+x+%3C+1

Answer 2

only valid for 0 ≤ x ≤ 1

Answer 3

dy/dx = 8x(1/2)(x-x^2)^(-1/2) (1 - 2x) + 8(x-x^2)^(1/2)
= 4(x-x^2)^(-1/2) [ x(1-2x) + 2(x-x^2) ]
= (4/√(x-x^2) ( x - 2x^2 + 2x - 2x^2)
= (4/√(x-x^2) (3x - 4x^2)

= 0 for a max/min

3x - 4x^2 = 0
x(3 - 4x) = 0
x = 0 ----> looks like it will yield a min
x = 3/4 --->looks like it will yield a max

f(0) = 0
f(3/4)
=8(3/4)√(3/4 - 9/16)
= 6√(3/16)
= 6√3/4 = 3√3/2 or 2.598

min is 0
max is aprr 2.60

http://www.wolframalpha.com/input/?i=maximum+of+8x%28√%28x+−+x%5E2%29%29+

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