Consider the function [tex]f(x)=8 x+2 x^{-1}[/tex]. For this function there are four important intervals: [tex](-\infty, A][/tex], [tex][A, B)[/tex], [tex](B, C][/tex], and [tex][C, \infty)[/tex] where [tex]A[/tex], and [tex]C[/tex] are the critical numbers and the function is not defined at [tex]B[/tex].
Find [tex]A[/tex], [tex]B[/tex], and [tex]C[/tex].
Answer (1)
Find the derivative of the function: f ′ ( x ) = 8 − x 2 2 .
Set the derivative to zero and solve for x to find the critical points: x = ± 2 1 .
Determine where the function is undefined: x = 0 .
Assign the values to A , B , and C such that A < B < C : A = − 2 1 , B = 0 , C = 2 1 .
A = − 2 1 , B = 0 , C = 2 1
Explanation
Problem Analysis We are given the function f ( x ) = 8 x + 2 x − 1 and asked to find the critical numbers A and C , and the point B where the function is not defined, such that the intervals are ( − ∞ , A ] , [ A , B ) , ( B , C ] , and [ C , ∞ ) .
Finding the Derivative First, we need to find the critical numbers by taking the derivative of f ( x ) and setting it equal to zero. The derivative of f ( x ) is f ′ ( x ) = d x d ( 8 x + 2 x − 1 ) = 8 − 2 x − 2 = 8 − x 2 2 .
Finding Critical Numbers Now, we set f ′ ( x ) = 0 and solve for x :
8 − x 2 2 = 0 ⇒ 8 = x 2 2 ⇒ x 2 = 8 2 = 4 1 ⇒ x = ± 2 1 . Thus, the critical numbers are x = − 2 1 and x = 2 1 .
Finding Undefined Point Next, we need to find where the function f ( x ) is not defined. Since f ( x ) = 8 x + x 2 , the function is not defined when x = 0 .
Assigning Values We are given that A , and C are the critical numbers and the function is not defined at B . Also, we know that A < B < C . Therefore, we have A = − 2 1 , B = 0 , and C = 2 1 .
Final Answer Thus, we have found that A = − 2 1 , B = 0 , and C = 2 1 .
Examples
Understanding critical points and intervals where a function is defined is crucial in optimization problems. For example, if you're designing a container to minimize surface area for a given volume, you'd use calculus to find critical dimensions. Knowing where the function is undefined helps avoid physically impossible solutions, ensuring your design is both mathematically optimal and practically feasible.
