a companys revenue (in thousands of dollars) from a campaign with budget x (thousands of dollars) is s(x) =…

a companys revenue (in thousands of dollars) from a campaign with budget x (thousands of dollars) is s(x) = -0.008x³ + 0.54x² + 260, 0 ≤ x ≤ 55. use interval notation where appropriate and enter values to 1 decimal place or as exact fractions. (a) find where s is increasing and decreasing. increasing on: decreasing on: (b) enter the x-value where s has its relative and absolute maximum: x = (c) determine the intervals of concavity. concave up on: concave down on: (d) find the inflection point (enter the x-value):

a companys revenue (in thousands of dollars) from a campaign with budget x (thousands of dollars) is s(x) = -0.008x³ + 0.54x² + 260, 0 ≤ x ≤ 55. use interval notation where appropriate and enter values to 1 decimal place or as exact fractions. (a) find where s is increasing and decreasing. increasing on: decreasing on: (b) enter the x-value where s has its relative and absolute maximum: x = (c) determine the intervals of concavity. concave up on: concave down on: (d) find the inflection point (enter the x-value):

Answer

Explanation:

Step1: Find the first derivative

Given ( S(x)=-0.008x^{3}+0.54x^{2}+260), the first derivative ( S^{\prime}(x)) is found using the power rule ((x^{n})^\prime = nx^{n - 1}). [ \begin{align*} S^{\prime}(x)&=-0.008\times3x^{2}+0.54\times2x+0\ &=- 0.024x^{2}+1.08x \end{align*} ] Set ( S^{\prime}(x)=0), so (-0.024x^{2}+1.08x = 0). Factor out (x): (x(-0.024x + 1.08)=0). Solving for (x), we get (x = 0) or (-0.024x+1.08 = 0). For (-0.024x+1.08 = 0), (x=\frac{1.08}{0.024}=45). We use test - points in the intervals ((0,45)) and ((45,55)). Let's take (x = 10) (in ((0,45))): (S^{\prime}(10)=-0.024\times10^{2}+1.08\times10=- 2.4 + 10.8=8.4>0). Let's take (x = 50) (in ((45,55))): (S^{\prime}(50)=-0.024\times50^{2}+1.08\times50=-60 + 54=-6<0).

Step2: Determine increasing and decreasing intervals

Since (S^{\prime}(x)>0) on the interval ((0,45)) and (S^{\prime}(x)<0) on the interval ((45,55)), the function (S(x)) is increasing on ((0,45)) and decreasing on ((45,55)).

Step3: Find the second derivative

The second derivative (S^{\prime\prime}(x)) is found by differentiating (S^{\prime}(x)=-0.024x^{2}+1.08x). Using the power rule, (S^{\prime\prime}(x)=-0.024\times2x+1.08=-0.048x + 1.08).

Step4: Find concavity and inflection point

Set (S^{\prime\prime}(x) = 0), so (-0.048x+1.08 = 0). Solving for (x), (x=\frac{1.08}{0.048}=22.5). Take a test - point in ((0,22.5)), say (x = 10): (S^{\prime\prime}(10)=-0.048\times10 + 1.08=-0.48+1.08 = 0.6>0). Take a test - point in ((22.5,55)), say (x = 30): (S^{\prime\prime}(30)=-0.048\times30+1.08=-1.44 + 1.08=-0.36<0).

Answer:

(a) Increasing on ((0,45)), decreasing on ((45,55)) (b) The function has a relative maximum at (x = 45) (since the function changes from increasing to decreasing at (x = 45)) (c) Concave up on ((0,22.5)), concave down on ((22.5,55)) (d) The inflection point is at (x = 22.5)