Is #f(x)=(x-3)/sqrt(x+3) # increasing or decreasing at #x=5 #?

Question

Is #f(x)=(x-3)/sqrt(x+3)   # increasing or decreasing at #x=5    #?

Aurora Alvarado · Accepted Answer

We start by differentiating.

Let #y = u^(1/2)# and #u = x + 3#. By the power rule, #y' = 1/2u^(-1/2) = 1/(2u^(1/2))# and #u' = 1#

By the chain rule:

#dy/dx= dy/(du) xx (du)/dx#

#dy/dx = 1/(2u^(1/2)) xx 1#

#dy/dx = 1/(2(x + 3)^(1/2))#

For #f'(x)#, we find the derivative by the quotient rule:

#f'(x) = (1sqrt(x+ 3) - (x - 3)1/(2(x + 3)^(1/2)))/(sqrt(x+ 3))^2#

#f'(x) = (sqrt(x + 3) - (x - 3)/(2(x + 3)^(1/2)))/(x + 3)#

#f'(x) = ((2sqrt(x + 3)sqrt(x+ 3) - x + 3)/(2(x + 3)^(1/2)))/(x+ 3)#

#f'(x) = ((2(x + 3) - x + 3)/(2(x+ 3)^(1/2)))/(x + 3)#

#f'(x) = ((2x + 6 - x + 3)/(2(x + 3)^(1/2)))/(x + 3)#

#f'(x) = (x + 9)/(2(x + 3)^(1/2)x + 3)#

#f'(x) = (x + 9)/(2(x + 3)^(3/2))#

Now, if the function is increasing at the point #x = a#, then #f'(a) > 0#. If the function is decreasing at #x= a#, then #f'(a) < 0#.

#f'(5) = (5 + 9)/(2(5 + 3)^(3/2))#

#f'(5) = 14/(2sqrt(512))#

#f'(5) = 14/(2(16)sqrt(2))#

#f'(5) = 14/(32sqrt(2)#

#f'(5) = 7/(16sqrt(2))#

What matters is that the above value is positive. Hence, #f(x)# is increasing at #x = 5#.

Hopefully this helps!

Christopher Allers · Answer

undefined To determine if $ f(x) = \frac{x - 3}{\sqrt{x + 3}} $ is increasing or decreasing at $ x = 5 $, we need to evaluate the derivative of $ f(x) $ and then determine its sign at $ x = 5 $.

The derivative of $ f(x) $ with respect to $ x $ is given by:

$$ f'(x) = \frac{d}{dx}\left(\frac{x - 3}{\sqrt{x + 3}}\right) $$

Using the quotient rule and the chain rule, we find:

$$ f'(x) = \frac{(x + 3)^\frac{-3}{2}}{2} - \frac{1}{2\sqrt{x + 3}} $$

Now, to determine the behavior of $ f(x) $ at $ x = 5 $, we substitute $ x = 5 $ into $ f'(x) $ and evaluate its sign.

$$ f'(5) = \frac{(5 + 3)^\frac{-3}{2}}{2} - \frac{1}{2\sqrt{5 + 3}} $$

$$ f'(5) = \frac{1}{2\sqrt{8}} - \frac{1}{2\sqrt{8}} = 0 $$

Since the derivative $ f'(5) = 0 $, we cannot determine the increasing or decreasing behavior of $ f(x) $ at $ x = 5 $ from this information alone. We may need to perform further analysis, such as using the second derivative test or examining the function's behavior in the surrounding interval.