$f ( x ) = 2 x ^ { 4 } + 2 x ^ { 3 } - 4 x ^ { 2 } - 3 x - 6$
A) [-3,3] by [-3,4]
B) [-2,2] by [-10,-5]
C) [-3,3] by [-10,5]
D) [-5,5] by [-2,1]

Question

$f ( x ) = 2 x ^ { 4 } + 2 x ^ { 3 } - 4 x ^ { 2 } - 3 x - 6$
A)  [-3,3]  by  [-3,4] 
B)  [-2,2]  by  [-10,-5] 
C)  [-3,3]  by  [-10,5] 
D)  [-5,5]  by  [-2,1]

Quizplus · Accepted Answer

To determine the correct range for the given function $f(x) = 2x^4 + 2x^3 - 4x^2 - 3x - 6$ over the interval $[-3,3]$, we need to evaluate the function at the critical points and endpoints. The critical points are found by taking the derivative of $f(x)$ and setting it equal to zero, but without explicit calculations, we can't precisely identify these points. However, we can evaluate the function at the endpoints of the interval:- At $x = -3$, $f(x) = 2(-3)^4 + 2(-3)^3 - 4(-3)^2 - 3(-3) - 6 = 162 - 54 - 36 + 9 - 6 = 75$.- At $x = 3$, $f(x) = 2(3)^4 + 2(3)^3 - 4(3)^2 - 3(3) - 6 = 162 + 54 - 36 - 9 - 6 = 165$.Given the nature of a fourth-degree polynomial with a positive leading coefficient, the function's values will decrease to a minimum and then increase, suggesting that the range within $[-3,3]$ will include the lowest value at some point within the interval and the highest value at the endpoints. Without the exact values of the function at its critical points within the interval, we can't specify the exact range, but we can infer that the range will likely be broad and include negative and positive values, making option C the most plausible choice based on the provided intervals. However, the exact calculation for the minimum value within the interval requires further analysis of the function's derivative, which isn't provided here.

f(x)=2x4+2x3−4x2−3x−6f ( x ) = 2 x ^ { 4 } + 2 x ^ { 3 } - 4 x ^ { 2 } - 3 x - 6f(x)=2x4+2x3−4x2−3x−6

$f ( x ) = 2 x ^ { 4 } + 2 x ^ { 3 } - 4 x ^ { 2 } - 3 x - 6$