Question

use the graph to complete each of the following parts.
(a) from the choices of values of ( x ) below, choose all at which ( f ) appears to have a vertical tangent line.
( square x = -3 ) ( square x = -2 ) ( square x = -1 ) ( square x = 0 )
( square x = 1 ) ( square x = 2 ) ( square x = 3 ) ( square ) none
(b) from the choices of values of ( x ) below, choose all at which ( f ) appears to have a \corner\ (or \kink\).
( square x = -3 ) ( square x = -2 ) ( square x = -1 ) ( square x = 0 )
( square x = 1 ) ( square x = 2 ) ( square x = 3 ) ( square ) none
(c) from the choices of values of ( x ) below, choose all at which ( f ) appears to have a discontinuity.
( square x = -3 ) ( square x = -2 ) ( square x = -1 ) ( square x = 0 )
( square x = 1 ) ( square x = 2 ) ( square x = 3 ) ( square ) none
(d) from the choices of values of ( x ) below, choose all at which ( f ) appears to be not differentiable.
( square x = -3 ) ( square x = -2 ) ( square x = -1 ) ( square x = 0 )
( square x = 1 ) ( square x = 2 ) ( square x = 3 ) ( square ) none

Explanation:

Step1: Identify vertical tangents

A vertical tangent appears where the graph curves sharply vertically, with the slope approaching $\pm\infty$. Looking at the graph, none of the $x$-values show this behavior.

Step2: Identify corners/kinks

A corner/kink is a point where the graph has a sudden change in direction (different left/right slopes). At $x=-2$, the graph switches from a upward-opening curve to a downward-sloping curve; at $x=2$, it switches from a upward-opening curve to a downward-sloping curve.

Step3: Identify discontinuities

Discontinuities occur where there is a break, hole, or jump. At $x=-3$, there is a hole with a different defined point; at $x=0$, there is a hole with a defined point below; at $x=3$, there is a hole with the graph continuing on either side.

Step4: Identify non-differentiable points

A function is non-differentiable at discontinuities, corners, and vertical tangents. Combine the points from (a), (b), (c): $x=-3, x=-2, x=0, x=2, x=3$.

Answer:

(a) None
(b) $x=-2$, $x=2$
(c) $x=-3$, $x=0$, $x=3$
(d) $x=-3$, $x=-2$, $x=0$, $x=2$, $x=3$