Consider continuous functions $f, g, h$, and $k$. Then complete the statements.

Function $h$ is two times the square of the difference of $x$ and 1.
$k(x)=x^4+2 x^2+8 x-4$

Select the correct answer from each drop-down.

The function that has the least minimum value is function $\square$.
The function that has the greatest minimum value is function $\square$.

Analyze g(x) from the table and find its minimum value is -5.
Analyze h(x) = 2(x-1)^2 and find its minimum value is 0 at x=1.
Analyze k(x) = x^4 + 2x^2 + 8x - 4, find its derivative k'(x) = 4x^3 + 4x + 8, and solve for critical points. Determine that x=-1 is a minimum using the second derivative test, and find k(-1) = -9.
Compare the minimum values: g(x) = -5, h(x) = 0, k(x) = -9. The least minimum is -9 (k), and the greatest minimum is 0 (h).

k , h ​
Explanation

Problem Analysis We are given three functions: g ( x ) defined by a table, h ( x ) = 2 ( x − 1 ) 2 , and k ( x ) = x 4 + 2 x 2 + 8 x − 4 . Our goal is to find the least and greatest minimum values among these functions.

Analyzing g(x) First, let's analyze the function g ( x ) using the provided table. The values of g ( x ) are: 76, 11, -4, -5, -4, 11. The minimum value of g ( x ) is -5, which occurs at x = 1 .

Analyzing h(x) Next, let's analyze the function h ( x ) = 2 ( x − 1 ) 2 . This is a parabola with a vertex at x = 1 . The minimum value of h ( x ) occurs at the vertex. h ( 1 ) = 2 ( 1 − 1 ) 2 = 2 ( 0 ) 2 = 0 . So, the minimum value of h ( x ) is 0.

Analyzing k(x) - Finding Critical Points Now, let's analyze the function k ( x ) = x 4 + 2 x 2 + 8 x − 4 . To find the minimum value, we need to find the critical points by taking the first derivative and setting it to zero. k ′ ( x ) = 4 x 3 + 4 x + 8 . Setting k ′ ( x ) = 0 , we get 4 x 3 + 4 x + 8 = 0 , which simplifies to x 3 + x + 2 = 0 .

Finding the Root of k'(x) We can observe that x = − 1 is a root of the equation x 3 + x + 2 = 0 because ( − 1 ) 3 + ( − 1 ) + 2 = − 1 − 1 + 2 = 0 . To confirm that this is the only real root, we can perform polynomial division to divide x 3 + x + 2 by ( x + 1 ) . This gives us x 2 − x + 2 . The discriminant of the quadratic x 2 − x + 2 is ( − 1 ) 2 − 4 ( 1 ) ( 2 ) = 1 − 8 = − 7 , which is negative. Therefore, there are no other real roots.

Second Derivative Test Now, we need to check if x = − 1 is a local minimum. We find the second derivative of k ( x ) : k ′′ ( x ) = 12 x 2 + 4 . Evaluating k ′′ ( − 1 ) , we get k ′′ ( − 1 ) = 12 ( − 1 ) 2 + 4 = 12 + 4 = 16 . Since 0"> k ′′ ( − 1 ) > 0 , x = − 1 is a local minimum.

Finding the Minimum Value of k(x) Now, we evaluate k ( x ) at x = − 1 to find the minimum value: k ( − 1 ) = ( − 1 ) 4 + 2 ( − 1 ) 2 + 8 ( − 1 ) − 4 = 1 + 2 − 8 − 4 = − 9 . So, the minimum value of k ( x ) is -9.

Comparing Minimum Values Comparing the minimum values of the three functions, we have: g ( x ) has a minimum of -5, h ( x ) has a minimum of 0, and k ( x ) has a minimum of -9. Therefore, the function with the least minimum value is k ( x ) with a minimum of -9, and the function with the greatest minimum value is h ( x ) with a minimum of 0.

Final Answer The function that has the least minimum value is function k . The function that has the greatest minimum value is function h .


Examples
Understanding minimum values of functions is crucial in optimization problems. For example, in manufacturing, a company might want to minimize the cost function to find the production level that results in the lowest cost. Similarly, in physics, finding the minimum potential energy helps determine the stable equilibrium of a system. By analyzing functions and their derivatives, we can identify these critical points and optimize various real-world processes.

Answered by GinnyAnswer | 2025-07-08

The least minimum value is for function k, which is -9. The greatest minimum value is for function h, which is 0. Therefore, the answers are k for the least minimum and h for the greatest minimum.
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Answered by Anonymous | 2025-07-11