A company's revenue (in thousands of dollars) from a campaign with budget $x$ (in thousands of dollars) is

$S(x) = -0.012x^3 + 0.54x^2 + 120, \quad 0 \leq x \leq 55$

Use interval notation where appropriate and enter values to 1 decimal place or as exact fractions.

AFind where $S$ is increasing and decreasing.

Increasing on:

Decreasing on:

BEnter the $x$-value where $S$ has its relative and absolute maximum:

$x =$

CDetermine the intervals of concavity.

Concave up on:

Concave down on:

DFind the inflection point (enter the $x$-value):

$x =$

Question

kaylagomez2008 kaylagomez2008

01-11-2026
Mathematics

Answered

A company's revenue (in thousands of dollars) from a campaign with budget $x$ (in thousands of dollars) is

$S(x) = -0.012x^3 + 0.54x^2 + 120, \quad 0 \leq x \leq 55$

Use interval notation where appropriate and enter values to 1 decimal place or as exact fractions.

AFind where $S$ is increasing and decreasing.

Increasing on:

Decreasing on:

BEnter the $x$-value where $S$ has its relative and absolute maximum:

$x =$

CDetermine the intervals of concavity.

Concave up on:

Concave down on:

DFind the inflection point (enter the $x$-value):

$x =$

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-01T01:03:50-04:00

The function is increasing on $(0, 30)$ and decreasing on $(30, 55)$ based on the sign of the first derivative.
The relative and absolute maximum occurs at $x = 30$.
The function is concave up on $(0, 15)$ and concave down on $(15, 55)$, with an inflection point at $x = 15$.
Final results:

AIncreasing on: $\boxed{(0, 30)}$, Decreasing on: $\boxed{(30, 55)}$

B$\boxed{30}$

CConcave up on: $\boxed{(0, 15)}$, Concave down on: $\boxed{(15, 55)}$

D$\boxed{15}$

Explanation

Problem Analysis

The revenue function is given by $S(x) = -0.012x^3 + 0.54x^2 + 120$ for the domain $x \in [0, 55]$. This is a cubic polynomial function, which is continuous and differentiable throughout its domain. We will analyze its behavior using the first and second derivatives to determine its monotonicity, extrema, and concavity.

Increasing and Decreasing Intervals

To find where the revenue is increasing or decreasing, we first compute the first derivative:
$S'(x) = \frac{d}{dx}(-0.012x^3 + 0.54x^2 + 120) = -0.036x^2 + 1.08x$.
Setting $S'(x) = 0$:
$-0.036x(x - 30) = 0$.
The critical points are $x = 0$ and $x = 30$.
Testing intervals within $[0, 55]$:

For $x \in (0, 30)$, pick $x=10$: $S'(10) = -0.036(100) + 1.08(10) = -3.6 + 10.8 = 7.2 > 0$, so the function is increasing.
For $x \in (30, 55)$, pick $x=40$: $S'(40) = -0.036(1600) + 1.08(40) = -57.6 + 43.2 = -14.4 < 0$, so the function is decreasing.

Thus, $S$ is increasing on $(0, 30)$ and decreasing on $(30, 55)$.

Finding the Maximum Revenue

We evaluate the function $S(x)$ at the critical points and boundaries:

$S(0) = 120$
$S(30) = -0.012(30^3) + 0.54(30^2) + 120 = -0.012(27000) + 0.54(900) + 120 = -324 + 486 + 120 = 282$
$S(55) = -0.012(55^3) + 0.54(55^2) + 120 = -0.012(166375) + 0.54(3025) + 120 = -1996.5 + 1633.5 + 120 = -243$

Comparing these values, the maximum revenue occurs at $x = 30$. This point is both a relative and an absolute maximum.

Concavity and Inflection Point

To find the concavity, we compute the second derivative:
$S''(x) = \frac{d}{dx}(-0.036x^2 + 1.08x) = -0.072x + 1.08$.
Setting $S''(x) = 0$ gives $0.072x = 1.08$, so $x = 15$.
Testing intervals:

For $x \in (0, 15)$, pick $x=1$: $S''(1) = -0.072(1) + 1.08 = 1.008 > 0$, so the function is concave up.
For $x \in (15, 55)$, pick $x=20$: $S''(20) = -0.072(20) + 1.08 = -1.44 + 1.08 = -0.36 < 0$, so the function is concave down.

The inflection point is at $x = 15$.

Final Answer Summary

The final results are:

AIncreasing on: $(0, 30)$, Decreasing on: $(30, 55)$.

BRelative and absolute maximum at $x = 30$.

CConcave up on: $(0, 15)$, Concave down on: $(15, 55)$.

DInflection point at $x = 15$.

Examples

A manager can use this revenue analysis to determine the optimal budget allocation for a marketing campaign. By identifying the budget value where the marginal revenue becomes zero (the maximum point of the revenue function), the company avoids overspending on a campaign that yields diminishing returns. This strategic use of calculus helps in maximizing profitability and efficient resource management.