Determine the range of values of [tex]$x$[/tex] for which [tex]$y=\frac{2 x-4}{x^2+5}$[/tex] is a decreasing function.
Answer (1)
Find the derivative of y = x 2 + 5 2 x − 4 using the quotient rule: d x d y = ( x 2 + 5 ) 2 − 2 x 2 + 8 x + 10 .
Set the derivative less than 0 to find when the function is decreasing: ( x 2 + 5 ) 2 − 2 x 2 + 8 x + 10 < 0 .
Simplify the inequality to 0"> x 2 − 4 x − 5 > 0 , then factor to get 0"> ( x − 5 ) ( x + 1 ) > 0 .
Solve the inequality to find the intervals where the function is decreasing: x < − 1 or 5"> x > 5 , which is ( − ∞ , − 1 ) ∪ ( 5 , ∞ ) . The final answer is 5}"> x < − 1 or x > 5 .
Explanation
Problem Analysis We are given the function y = x 2 + 5 2 x − 4 and we want to find the range of x values for which the function is decreasing. A function is decreasing when its derivative is negative, so we need to find the derivative of y with respect to x and determine when it is less than 0.
Finding the Derivative To find the derivative d x d y , we use the quotient rule. If y = v u , then d x d y = v 2 v d x d u − u d x d v . In this case, u = 2 x − 4 and v = x 2 + 5 . So, d x d u = 2 and d x d v = 2 x . Applying the quotient rule, we get: d x d y = ( x 2 + 5 ) 2 ( x 2 + 5 ) ( 2 ) − ( 2 x − 4 ) ( 2 x ) Simplifying the numerator: d x d y = ( x 2 + 5 ) 2 2 x 2 + 10 − ( 4 x 2 − 8 x ) = ( x 2 + 5 ) 2 2 x 2 + 10 − 4 x 2 + 8 x = ( x 2 + 5 ) 2 − 2 x 2 + 8 x + 10 So, d x d y = ( x 2 + 5 ) 2 − 2 x 2 + 8 x + 10 .
Setting up the Inequality We want to find where d x d y < 0 . Since the denominator ( x 2 + 5 ) 2 is always positive, we only need to consider the numerator: − 2 x 2 + 8 x + 10 < 0 Divide by -2 (and reverse the inequality): 0"> x 2 − 4 x − 5 > 0 Factoring the quadratic: 0"> ( x − 5 ) ( x + 1 ) > 0 To solve this inequality, we consider the intervals determined by the roots x = − 1 and x = 5 .
Solving the Inequality We test the intervals:
x < − 1 : Choose x = − 2 . Then 0"> ( − 2 − 5 ) ( − 2 + 1 ) = ( − 7 ) ( − 1 ) = 7 > 0 . So the inequality holds for x < − 1 .
− 1 < x < 5 : Choose x = 0 . Then ( 0 − 5 ) ( 0 + 1 ) = ( − 5 ) ( 1 ) = − 5 < 0 . So the inequality does not hold for − 1 < x < 5 .
5"> x > 5 : Choose x = 6 . Then 0"> ( 6 − 5 ) ( 6 + 1 ) = ( 1 ) ( 7 ) = 7 > 0 . So the inequality holds for 5"> x > 5 .
Thus, the solution to the inequality is x < − 1 or 5"> x > 5 .
Final Answer Therefore, the function y = x 2 + 5 2 x − 4 is decreasing when x < − 1 or 5"> x > 5 . In interval notation, this is ( − ∞ , − 1 ) ∪ ( 5 , ∞ ) .
Conclusion The range of values of x for which the function is decreasing is x < − 1 or 5"> x > 5 .
Examples
Understanding when a function is increasing or decreasing is crucial in many real-world applications. For example, in economics, a cost function might be modeled as y = x 2 + 5 2 x − 4 , where x represents the quantity of goods produced. Determining when this function is decreasing helps businesses identify the production levels at which the cost per unit decreases, leading to more efficient operations and higher profits. Similarly, in physics, this could represent the velocity of an object, and knowing when the velocity is decreasing (i.e., decelerating) is essential for analyzing motion.
