What are the absolute extrema of #f(x)=x-sqrt(5x-2) in(2,5)#?

1 Answer
Jul 13, 2018

Answer:

There are no absolute extrema in the interval #(2, 5)#

Explanation:

Given: #f(x) = x - sqrt(5x - 2) in (2, 5)#

To find absolute extrema we need to find the first derivative and perform the first derivative test to find any minimum or maximums and then find the #y# values of the end points and compare them.

Find the first derivative:

#f(x) = x - (5x - 2)^(1/2)#

#f'(x) = 1 - 1/2( 5x - 2)^(-1/2)(5)#

#f'(x) = 1 - 5/(2sqrt(5x - 2))#

Find critical value(s) #f'(x) = 0#:

#1 - 5/(2sqrt(5x - 2)) = 0#

#1 = 5/(2sqrt(5x - 2))#

#2sqrt(5x - 2) = 5#

#sqrt(5x - 2) = 5/2#

Square both sides: #5x - 2 = +- 25/4#

Since the domain of the function is limited by the radical:

#5x - 2 >= 0; " "x >= 2/5#

We only need to look at the positive answer:

#5x - 2 = + 25/4#

#5x = 2/1 *4/4 + 25/4 = 33/4#

#x = 33/4 * 1/5 = 33/20 ~~1.65#

Since this critical point is #< 2#, we can ignore it.

This means the absolute extrema are at the endpoints, but the endpoints are not included in the interval.