Answered

Which of the two functions below has the largest maximum [tex]\( y \)[/tex]-value?

[tex]\( f(x) = -x^4 - 2 \)[/tex]
[tex]\( g(x) = -3x^3 + 2 \)[/tex]

A. [tex]\( f(x) \)[/tex]
B. There is not enough information to determine
C. The extreme maximum [tex]\( y \)[/tex]-value for both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is [tex]\(\infty\)[/tex]
D. [tex]\( g(x) \)[/tex]



Answer :

To determine which of the two functions [tex]\( f(x) = -x^4 - 2 \)[/tex] and [tex]\( g(x) = -3x^3 + 2 \)[/tex] has the largest maximum [tex]\( y \)[/tex]-value, we need to find their critical points and evaluate the functions at those points.

### Finding Critical Points
First, we find the critical points by taking the derivative of each function and setting it equal to zero.

1. For [tex]\( f(x) = -x^4 - 2 \)[/tex]:

The derivative, [tex]\( f'(x) \)[/tex], is calculated as:
[tex]\[ f'(x) = \frac{d}{dx} (-x^4 - 2) = -4x^3 \][/tex]

Setting [tex]\( f'(x) = 0 \)[/tex]:
[tex]\[ -4x^3 = 0 \implies x = 0 \][/tex]

2. For [tex]\( g(x) = -3x^3 + 2 \)[/tex]:

The derivative, [tex]\( g'(x) \)[/tex], is calculated as:
[tex]\[ g'(x) = \frac{d}{dx} (-3x^3 + 2) = -9x^2 \][/tex]

Setting [tex]\( g'(x) = 0 \)[/tex]:
[tex]\[ -9x^2 = 0 \implies x = 0 \][/tex]

### Evaluating the Functions at the Critical Points
Next, evaluate each function at its critical point to determine the maximum [tex]\( y \)[/tex]-value.

1. For [tex]\( f(x) \)[/tex]:

Substitute [tex]\( x = 0 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = -0^4 - 2 = -2 \][/tex]

So, the maximum [tex]\( y \)[/tex]-value for [tex]\( f(x) \)[/tex] is [tex]\( -2 \)[/tex].

2. For [tex]\( g(x) \)[/tex]:

Substitute [tex]\( x = 0 \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(0) = -3(0)^3 + 2 = 2 \][/tex]

So, the maximum [tex]\( y \)[/tex]-value for [tex]\( g(x) \)[/tex] is [tex]\( 2 \)[/tex].

### Conclusion

Comparing the maximum [tex]\( y \)[/tex]-values of both functions:
- The maximum [tex]\( y \)[/tex]-value for [tex]\( f(x) \)[/tex] is [tex]\( -2 \)[/tex].
- The maximum [tex]\( y \)[/tex]-value for [tex]\( g(x) \)[/tex] is [tex]\( 2 \)[/tex].

Hence, the function [tex]\( g(x) \)[/tex] has the largest maximum [tex]\( y \)[/tex]-value.

Therefore, the answer is:
D. [tex]\( g(x) \)[/tex]