Question

identify the inflection points and local maxima and minima of the graphed function. identify the open intervals on which the function is differentiable and is concave up and concave down.
answer in interval notation.)
b. the function is never concave up.
find the open interval(s) on which the function is differentiable and is concave down. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the interval(s) is/are
(simplify your answer. use a comma to separate answers as needed. type your answer in interval notation.)
b. the function is never concave down.

Explanation:

Step1: Find the first and second derivatives

Given \( y=\frac{x^{3}}{3}-x^{2}-3x \).
The first derivative \( y^\prime=\frac{d}{dx}(\frac{x^{3}}{3}-x^{2}-3x)=x^{2}-2x - 3\).
The second derivative \( y^{\prime\prime}=\frac{d}{dx}(x^{2}-2x - 3)=2x-2\).

Step2: Find the inflection point

Set \( y^{\prime\prime}=0\), so \( 2x - 2=0\).
Solving for \( x\): \( 2x=2\), \( x = 1\).

Step3: Determine concavity

For concavity, we test intervals based on the inflection - point \( x = 1\).

• When \( x<1\) (e.g., \( x = 0\)), \( y^{\prime\prime}(0)=2\times0 - 2=-2<0\).
• When \( x>1\) (e.g., \( x = 2\)), \( y^{\prime\prime}(2)=2\times2 - 2=2>0\).

Since the function \( y=\frac{x^{3}}{3}-x^{2}-3x\) is a polynomial, it is differentiable for all \( x\in(-\infty,\infty)\).

Answer:

The interval on which the function is differentiable and concave down is \( (-\infty,1)\).