Question

graph the function ( y = x ^ { 2 } - 2 x - 8 ) by identifying the domain and any symmetries, finding the derivatives ( y ^ { prime } ) and ( y ^ { prime prime } ), finding the critical points and identifying the functions behavior at each one, finding where the curve is increasing and where it is decreasing, finding the points of inflection, determining the concavity of the curve, identifying any asymptotes, and plotting any key points such as intercepts, critical points, and inflection points. then find coordinates of absolute extreme points, if any.
a. the critical point(s) occur(s) at ( x = 1 ).
(use a comma to separate answers as needed.)
b. there are no critical points.
identify any local minima. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the local minimum/minima is/are located at
(type an ordered pair. use a comma to separate answers as needed.)
b. there are no local minima.

Explanation:

Step1: Find the first derivative

Given \(y = x^{2}-2x - 8\), using the power rule \((x^n)^\prime=nx^{n - 1}\), we have \(y^\prime=\frac{d}{dx}(x^{2}-2x - 8)=2x-2\).

Step2: Find the critical points

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

$$
LATEXBLOCK0
$$

Step3: Find the second derivative

Differentiate \(y^\prime = 2x - 2\) with respect to \(x\). Using the power rule, \(y^{\prime\prime}=\frac{d}{dx}(2x - 2)=2\).
Since \(y^{\prime\prime}(1)=2>0\), by the second - derivative test, the function has a local minimum at \(x = 1\).

Step4: Find the \(y\) - value of the local minimum

Substitute \(x = 1\) into the original function \(y=x^{2}-2x - 8\).

$$
LATEXBLOCK1
$$

Answer:

A. The local minimum/minima is/are located at \((1,-9)\)