Question

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):

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 \((x^{n})^\prime = nx^{n - 1}\).

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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 \(x = 22.5\)