The polynomial function [tex]$F(x)=2 x^2+8 x-7$[/tex] has a critical point at which of the following [tex]$x$[/tex]-values?
A. [tex]$x=0$[/tex]
B. [tex]$x=2$[/tex]
C. [tex]$x=-7$[/tex]
D. [tex]$x=-2$[/tex]
Answer (1)
Find the derivative of the function: F ′ ( x ) = 4 x + 8 .
Set the derivative equal to zero: 4 x + 8 = 0 .
Solve for x : x = − 2 .
The critical point occurs at x = − 2 .
Explanation
Problem Analysis We are given the polynomial function F ( x ) = 2 x 2 + 8 x − 7 and asked to find the x -value where it has a critical point. Critical points occur where the derivative of the function is equal to zero or undefined. In this case, since F ( x ) is a polynomial, its derivative is defined for all x .
Finding the Derivative To find the critical point, we first need to find the derivative of F ( x ) with respect to x . Using the power rule, we have: F ′ ( x ) = d x d ( 2 x 2 + 8 x − 7 ) = 4 x + 8
Setting the Derivative to Zero Now, we set the derivative equal to zero and solve for x :
4 x + 8 = 0
Isolating x Subtracting 8 from both sides gives: 4 x = − 8
Solving for x Dividing both sides by 4, we find: x = − 2
Final Answer Therefore, the polynomial function F ( x ) = 2 x 2 + 8 x − 7 has a critical point at x = − 2 .
Examples
In physics, understanding critical points is essential when analyzing potential energy functions. For instance, the potential energy of a system might be described by a quadratic function similar to the one in this problem. Finding the critical point of this potential energy function allows us to determine the point of stable equilibrium, which is crucial for understanding the system's behavior. In economics, critical points can help determine points of maximum profit or minimum cost in a cost-benefit analysis.
