Over which interval is the graph of [tex]f(x)=-x^2+3 x+[/tex] 8 increasing?
A. ( [tex]-\infty, 1.5[/tex] )
B. ( [tex]-\infty, 10.25[/tex] )
C. ([tex]1.5, \infty[/tex])
D. ([tex]10.25, \infty[/tex])
Answer (1)
Find the derivative of the function: f ′ ( x ) = − 2 x + 3 .
Set the derivative greater than zero: 0"> − 2 x + 3 > 0 .
Solve the inequality for x : x < 1.5 .
The function is increasing on the interval: ( − ∞ , 1.5 ) .
Explanation
Problem Analysis We are given the function f ( x ) = − x 2 + 3 x + 8 and asked to find the interval over which the graph of f ( x ) is increasing.
Objective To find the interval where f ( x ) is increasing, we need to find the derivative f ′ ( x ) and determine where 0"> f ′ ( x ) > 0 .
Finding the Derivative First, we find the derivative of f ( x ) :
f ′ ( x ) = d x d ( − x 2 + 3 x + 8 ) = − 2 x + 3
Solving the Inequality Next, we set 0"> f ′ ( x ) > 0 and solve for x :
0"> − 2 x + 3 > 0 2x"> 3 > 2 x x < 2 3 x < 1.5
Determining the Interval Therefore, the function f ( x ) is increasing on the interval ( − ∞ , 1.5 ) .
Final Answer The graph of f ( x ) is increasing over the interval ( − ∞ , 1.5 ) .
Examples
Understanding where a function is increasing or decreasing is crucial in many real-world applications. For example, in economics, a company might want to know when their profit function is increasing to determine the optimal production level. Similarly, in physics, understanding when the velocity of an object is increasing helps to analyze its motion. In engineering, this concept is used to optimize designs and processes.
