A function is matched below against an interval where it is...

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched? Interval Function (A) $(-\infty, \infty)$ $x^{3}-3 x^{2}+3 x+3$ (B) $[2, \infty)$ $2 x^{3} 3 x^{2}-12 x+6$ (C) $\left(-\infty, \frac{1}{3}\right]$ $3 x^{2}-2 x+1$ (D) $(-\infty,-4]$ $x^{3}+6 x^{2}+6$

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?
Interval Function
(A) $(-\infty, \infty)$
$x^{3}-3 x^{2}+3 x+3$
(B) $[2, \infty)$
$2 x^{3} 3 x^{2}-12 x+6$
(C) $\left(-\infty, \frac{1}{3}\right]$
$3 x^{2}-2 x+1$
(D) $(-\infty,-4]$
$x^{3}+6 x^{2}+6$

Key Concepts

Monotonicity

Monotonicity in calculus refers to the behavior of a function in which it either never decreases or never increases over an interval. An increasing function is one where, for any two points in its domain with one less than the other, the function’s value is less than or equal to the value at the later point. This concept is fundamental when analyzing whether a function behaves steadily upward over a given interval without any dips or plateaus.

First Derivative Test

The first derivative test is used to determine intervals where a function is increasing or decreasing. By computing the derivative of a function and analyzing its sign over particular intervals, one can ascertain if the function is consistently rising. If the first derivative is positive throughout an interval, the function is increasing there, which is crucial for matching the function with its increasing interval.

Critical Points

Critical points are values in the domain of a function where its first derivative is either zero or undefined. These points are significant because they often signal a change in the behavior of the function from increasing to decreasing, or vice versa. Identifying and analyzing these points helps in partitioning the domain into intervals where the function might be monotonic.

Derivative Analysis of Polynomials

Since the functions in question are polynomials, derivative analysis becomes a straightforward yet powerful tool. Polynomials have continuous derivatives, and their derivatives are easier to compute. By analyzing the sign of the polynomial's derivative, one can determine the intervals where the function is increasing or decreasing, making this concept essential for solving problems related to function behavior over specified intervals.

Transcript

Please give Ace some feedback