Over which interval is the graph of [tex]f(x)=\frac{1}{2} x^2+5 x+[/tex] 6 increasing?
A. ( [tex]-6.5, \infty[/tex] )
B. ( [tex]-5, \infty[/tex])
C. ( [tex]-\infty,-5[/tex])
D. ( [tex]-\infty,-6.5[/tex])
Answer (1)
Find the derivative of the function: f ′ ( x ) = x + 5 .
Set the derivative greater than zero: 0"> x + 5 > 0 .
Solve for x : -5"> x > − 5 .
The function is increasing on the interval: ( − 5 , ∞ ) .
Explanation
Problem Analysis We are given the function f ( x ) = f r a c 1 2 x 2 + 5 x + 6 and we want to find the interval where the function is increasing. A function is increasing when its derivative is positive.
Finding the Derivative First, we need to find the derivative of the function f ( x ) . Using the power rule, we have:
f ′ ( x ) = d x d ( f r a c 1 2 x 2 + 5 x + 6 ) = 2 1 ( 2 x ) + 5 = x + 5
Finding the Interval Now, we need to find where the derivative is greater than zero, i.e., 0"> f ′ ( x ) > 0 :
0"> x + 5 > 0
Subtracting 5 from both sides, we get:
-5"> x > − 5
Determining the Interval This means the function is increasing for all x greater than − 5 . In interval notation, this is ( − 5 , ∞ ) .
Final Answer Therefore, the graph of f ( x ) = 2 1 x 2 + 5 x + 6 is increasing over the interval ( − 5 , ∞ ) .
Examples
Understanding where a function is increasing or decreasing is crucial in many real-world applications. For example, if you're analyzing the growth of a population over time, knowing the intervals where the growth rate (the derivative) is positive tells you when the population is increasing. Similarly, in economics, you might analyze the profit function of a company to determine when production levels lead to increasing profits. This concept is also used in physics to analyze the motion of objects, determining when their velocity (the derivative of position) is positive, indicating movement in a certain direction.
