Question

sketch the following curve, indicating all relative extreme points and inflection points
$y = \frac{1}{3}x^{3}-4x^{2}+15x - 5$
the relative extreme points are $(3,13),(5,\frac{35}{3})$
(type an ordered pair. simplify your answer. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed. )
the inflection points are $(4,\frac{37}{3})$
(type an ordered pair. simplify your answer. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed )
choose the correct graph of $y = \frac{1}{3}x^{3}-4x^{2}+15x - 5$.
o a.
o b.
o c.
o d.

Explanation:

Step1: Analyze relative - extreme points

Relative extreme points occur where $y' = 0$. First, find the derivative of $y=\frac{1}{3}x^{3}-4x^{2}+15x - 5$. Using the power rule $(x^n)'=nx^{n - 1}$, we have $y'=x^{2}-8x + 15$. Set $y'=0$, then $x^{2}-8x + 15=(x - 3)(x - 5)=0$. Solving for $x$ gives $x = 3$ and $x = 5$. Substitute $x = 3$ into $y$: $y=\frac{1}{3}(3)^{3}-4(3)^{2}+15(3)-5=9 - 36+45 - 5 = 13$. Substitute $x = 5$ into $y$: $y=\frac{1}{3}(5)^{3}-4(5)^{2}+15(5)-5=\frac{125}{3}-100 + 75-5=\frac{125}{3}-30=\frac{125 - 90}{3}=\frac{35}{3}$. So the relative - extreme points are $(3,13)$ and $(5,\frac{35}{3})$.

Step2: Analyze inflection points

Inflection points occur where $y'' = 0$. Differentiate $y'=x^{2}-8x + 15$ to get $y''=2x-8$. Set $y'' = 0$, then $2x-8 = 0$, which gives $x = 4$. Substitute $x = 4$ into $y$: $y=\frac{1}{3}(4)^{3}-4(4)^{2}+15(4)-5=\frac{64}{3}-64 + 60-5=\frac{64}{3}-9=\frac{64 - 27}{3}=\frac{37}{3}$. So the inflection point is $(4,\frac{37}{3})$.

Step3: Analyze the graph

We know that the function is a cubic function $y=\frac{1}{3}x^{3}-4x^{2}+15x - 5$ with a positive leading coefficient $\frac{1}{3}$. At $x = 3$, we have a local maximum $(3,13)$ and at $x = 5$ we have a local minimum $(5,\frac{35}{3})$, and an inflection point at $(4,\frac{37}{3})$. By checking the behavior of the function around these points, we can determine the correct graph.

Answer:

We need to visually inspect the given graphs (not shown in detail here) and choose the one that has a local maximum at $(3,13)$, a local minimum at $(5,\frac{35}{3})$ and an inflection point at $(4,\frac{37}{3})$. Without the actual visual options for A, B, C, and D, we can't give a specific letter - answer, but the process to find the correct graph is as described above. If you provide the details of the graphs, we can further determine the correct choice.