Question

for the following function, a) give the coordinates of any critical points and classify each point as a relative maximum, a relative minimum, or neither, b) identify intervals where the function is increasing or decreasing, c) give the coordinates of any points of inflection, d) identify intervals where the function is concave up or concave down, and e) sketch the graph
$g(x)=x^{3}-12x^{2}+45x + 8$

Explanation:

Step1: Find the first derivative

The function is \(g(x)=x^{3}-12x^{2}+45x + 8\).
Using the power rule \((x^{n})^\prime=nx^{n - 1}\), we have \(g^\prime(x)=3x^{2}-24x + 45\).
Factor out 3: \(g^\prime(x)=3(x^{2}-8x + 15)=3(x - 3)(x - 5)\).
Set \(g^\prime(x)=0\), then \(3(x - 3)(x - 5)=0\).
Solving for \(x\), we get \(x = 3\) and \(x = 5\).
When \(x = 3\), \(g(3)=3^{3}-12\times3^{2}+45\times3 + 8=27-108 + 135+8=62\).
When \(x = 5\), \(g(5)=5^{3}-12\times5^{2}+45\times5 + 8=125-300 + 225+8=58\).
To classify the critical points, we use the second - derivative test.
Find the second derivative: \(g^{\prime\prime}(x)=6x-24\).
When \(x = 3\), \(g^{\prime\prime}(3)=6\times3-24=-6<0\), so \((3,62)\) is a relative maximum.
When \(x = 5\), \(g^{\prime\prime}(5)=6\times5-24 = 6>0\), so \((5,58)\) is a relative minimum.

Step2: Determine the intervals of increase and decrease

Test intervals for \(g^\prime(x)\):

• For \(x<3\), let \(x = 0\), \(g^\prime(0)=3(0 - 3)(0 - 5)=45>0\). So \(g(x)\) is increasing on \((-\infty,3)\).
• For \(3
• For \(x>5\), let \(x = 6\), \(g^\prime(6)=3(6 - 3)(6 - 5)=9>0\). So \(g(x)\) is increasing on \((5,\infty)\).

Step3: Find the points of inflection

Set \(g^{\prime\prime}(x)=0\), \(6x-24 = 0\), solving for \(x\) gives \(x = 4\).
When \(x = 4\), \(g(4)=4^{3}-12\times4^{2}+45\times4 + 8=64-192+180 + 8=60\). So the point of inflection is \((4,60)\).

Step4: Determine the intervals of concavity

Test intervals for \(g^{\prime\prime}(x)\):

• For \(x<4\), let \(x = 3\), \(g^{\prime\prime}(3)=6\times3-24=-6<0\). So \(g(x)\) is concave down on \((-\infty,4)\).
• For \(x>4\), let \(x = 5\), \(g^{\prime\prime}(5)=6\times5-24 = 6>0\). So \(g(x)\) is concave up on \((4,\infty)\).

Answer:

a) Critical points: \((3,62)\) (relative maximum), \((5,58)\) (relative minimum).
b) Increasing intervals: \((-\infty,3)\cup(5,\infty)\); Decreasing interval: \((3,5)\).
c) Point of inflection: \((4,60)\).
d) Concave down interval: \((-\infty,4)\); Concave up interval: \((4,\infty)\).