$f ( x ) = x ^ { 5 } + 5 x ^ { 4 } - x ^ { 3 } - 10 x ^ { 2 } + 3 x + 10$
A) [-10,10] by [-10,10]
B) [-6,6] by [-10,200]
C) [-10,10] by [-100,100]
D) [-3,3] by [-10,30]

Question

$f ( x ) = x ^ { 5 } + 5 x ^ { 4 } - x ^ { 3 } - 10 x ^ { 2 } + 3 x + 10$
A)  [-10,10]  by  [-10,10] 
B)  [-6,6]  by  [-10,200] 
C)  [-10,10]  by  [-100,100] 
D)  [-3,3]  by  [-10,30]

Quizplus · Accepted Answer

The correct choice is determined by evaluating the function's behavior over the specified intervals. The function $f(x) = x^5 + 5x^4 - x^3 - 10x^2 + 3x + 10$ is a fifth-degree polynomial, which means its values can vary significantly even within small intervals. The choice [-6,6] by [-10,200] is the most reasonable because, given the degree of the polynomial and the coefficients, the function's values can increase rapidly, especially for positive values of $x$, leading to a wide range of outputs that could easily exceed 100 in magnitude for inputs within [-6,6]. Other intervals either underestimate the range of $f(x)$ or overestimate the domain's necessary breadth to achieve such a range.

f(x)=x5+5x4−x3−10x2+3x+10f ( x ) = x ^ { 5 } + 5 x ^ { 4 } - x ^ { 3 } - 10 x ^ { 2 } + 3 x + 10f(x)=x5+5x4−x3−10x2+3x+10

$f ( x ) = x ^ { 5 } + 5 x ^ { 4 } - x ^ { 3 } - 10 x ^ { 2 } + 3 x + 10$