$\frac{21 x}{\sqrt{(x+2)}}$
Answer (1)
The domain of the expression is -2"> x > − 2 .
The first derivative is 2 ( x + 2 ) 3/2 21 ( x + 4 ) , which is always positive for -2"> x > − 2 , so the function is always increasing.
The second derivative is 4 ( x + 2 ) 5/2 − 21 ( x + 8 ) , which is always negative for -2"> x > − 2 , so the function is always concave down.
There are no critical points or inflection points. -2, \text{ Increasing, Concave Down, No Critical Points, No Inflection Points}}"> Domain: x > − 2 , Increasing, Concave Down, No Critical Points, No Inflection Points
Explanation
Determine the Domain We are given the expression x + 2 21 x and we want to analyze it. First, we need to determine the domain of the expression. Since we have a square root in the denominator, we must have 0"> x + 2 > 0 , which means -2"> x > − 2 . So the domain of the expression is ( − 2 , ∞ ) .
Find the First Derivative Next, let's find the derivative of the expression with respect to x . Using the quotient rule, we have: d x d ( x + 2 21 x ) = x + 2 21 x + 2 − 21 x ⋅ 2 x + 2 1 = ( x + 2 ) 3/2 21 ( x + 2 ) − 2 21 x = ( x + 2 ) 3/2 2 21 x + 42 = 2 ( x + 2 ) 3/2 21 ( x + 4 )
Find Critical Points Now, let's find the critical points by setting the derivative equal to zero. We have 2 ( x + 2 ) 3/2 21 ( x + 4 ) = 0 , which implies x + 4 = 0 , so x = − 4 . However, x = − 4 is not in the domain of the expression, so there are no critical points in the domain.
Analyze the Sign of the First Derivative Let's analyze the sign of the derivative. Since -2"> x > − 2 , we have 0"> ( x + 2 ) 3/2 > 0 . Also, 0"> x + 4 > 0 for -4"> x > − 4 , so 0"> x + 4 > 0 for all -2"> x > − 2 . Therefore, the derivative is always positive for -2"> x > − 2 , which means the expression is always increasing on its domain.
Find the Second Derivative Now, let's find the second derivative of the expression. We have: d x 2 d 2 ( x + 2 21 x ) = d x d ( 2 ( x + 2 ) 3/2 21 ( x + 4 ) ) = 2 21 ⋅ x + 2 ) 3 ( x + 2 ) 3/2 − ( x + 4 ) ⋅ 2 3 ( x + 2 ) 1/2 = 2 21 ⋅ ( x + 2 ) 5/2 ( x + 2 ) − 2 3 ( x + 4 ) = 2 21 ⋅ ( x + 2 ) 5/2 − 2 1 x − 4 = 4 ( x + 2 ) 5/2 − 21 ( x + 8 )
Analyze the Sign of the Second Derivative Now, let's analyze the sign of the second derivative. Since -2"> x > − 2 , we have 0"> ( x + 2 ) 5/2 > 0 . Also, 0"> x + 8 > 0 for -8"> x > − 8 , so 0"> x + 8 > 0 for all -2"> x > − 2 . Therefore, the second derivative is always negative for -2"> x > − 2 , which means the expression is always concave down on its domain.
Find Inflection Points Finally, let's find any inflection points by setting the second derivative equal to zero. We have 4 ( x + 2 ) 5/2 − 21 ( x + 8 ) = 0 , which implies x + 8 = 0 , so x = − 8 . However, x = − 8 is not in the domain of the expression, so there are no inflection points.
Summary In summary, the expression x + 2 21 x has a domain of ( − 2 , ∞ ) , is always increasing, and is always concave down. There are no critical points or inflection points.
Examples
Understanding the behavior of functions like this is crucial in fields like physics and engineering. For example, if 'x' represents time and the function represents the velocity of an object, knowing the domain, intervals of increase, and concavity helps predict the object's motion over time. Similarly, in economics, if 'x' represents investment and the function represents profit, analyzing these properties can help optimize investment strategies.
