Over what interval is the graph of [tex]f(x)=-(x+8)^2-1[/tex] decreasing?
A. [tex](-8, \infty)[/tex]
B. [tex](8, \infty)[/tex]
C. [tex](-\infty, 8)[/tex]
D. [tex](-\infty,-8)[/tex]
Answer (2)
Find the derivative of the function: f ′ ( x ) = − 2 x − 16 .
Find the critical point by setting the derivative to zero: x = − 8 .
Analyze the sign of the derivative in the intervals ( − ∞ , − 8 ) and ( − 8 , ∞ ) .
The function is decreasing on the interval ( − 8 , ∞ ) . The final answer is ( − 8 , ∞ ) .
Explanation
Problem Analysis We are given the function f ( x ) = − ( x + 8 ) 2 − 1 and asked to find the interval where the graph of f ( x ) is decreasing. To do this, we need to find the derivative of f ( x ) and determine where the derivative is negative.
Finding the Derivative First, we find the derivative of f ( x ) with respect to x . Using the chain rule, we have: f ′ ( x ) = − 2 ( x + 8 ) ( 1 ) − 0 = − 2 ( x + 8 ) = − 2 x − 16
Finding Critical Points Next, we find the critical points by setting the derivative equal to zero and solving for x :
− 2 x − 16 = 0 − 2 x = 16 x = − 8
Analyzing the Sign of the Derivative Now, we analyze the sign of the derivative in the intervals determined by the critical point x = − 8 . We have two intervals to consider: ( − ∞ , − 8 ) and ( − 8 , ∞ ) .
For the interval ( − ∞ , − 8 ) , let's test x = − 9 :
0"> f ′ ( − 9 ) = − 2 ( − 9 ) − 16 = 18 − 16 = 2 > 0 Since the derivative is positive in this interval, the function is increasing.
For the interval ( − 8 , ∞ ) , let's test x = − 7 :
f ′ ( − 7 ) = − 2 ( − 7 ) − 16 = 14 − 16 = − 2 < 0 Since the derivative is negative in this interval, the function is decreasing.
Determining the Interval of Decreasing Function Therefore, the graph of f ( x ) is decreasing on the interval ( − 8 , ∞ ) .
Examples
Understanding where a function is increasing or decreasing is useful in many real-world applications. For example, if f ( x ) represents the profit of a company as a function of the number of products sold, then knowing where f ( x ) is decreasing tells us the range of sales where the company is losing money. Similarly, in physics, if f ( x ) represents the height of a ball thrown in the air as a function of time, then knowing where f ( x ) is decreasing tells us when the ball is falling back to the ground. This concept is also used in optimization problems, where we want to find the maximum or minimum value of a function.
The function f ( x ) = − ( x + 8 ) 2 − 1 is decreasing on the interval ( − 8 , ∞ ) . This conclusion is reached by analyzing the derivative of the function and determining the sign of the derivative in the relevant intervals. Therefore, the correct option is A . ( − 8 , ∞ ) .
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