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In Exercises 11 and $12,$ mark each statement True or False. Justify each answer.a. The cubic Bézier curve is based on four control points.b. Given a quadratic Bézier curve $\mathbf{x}(t)$ with control points $\mathbf{p}_{0}, \mathbf{p}_{1},$ and $\mathbf{p}_{2},$ the directed line segment $\mathbf{p}_{1}-\mathbf{p}_{0}$ (from $\mathbf{p}_{0}$ to $\mathbf{p}_{1} )$ is the tangent vector to the curve at $\mathbf{p}_{0}$ .c. When two quadratic Bézier curves with control points $\left\{\mathbf{p}_{0}, \mathbf{p}_{1}, \mathbf{p}_{2}\right\}$ and $\left\{\mathbf{p}_{2}, \mathbf{p}_{3}, \mathbf{p}_{4}\right\}$ are joined at $\mathbf{p}_{2},$ the combined Bézier curve will have $C^{1}$ continuity at $\mathbf{p}_{2}$ if $\mathbf{p}_{2}$ is the midpoint of the line segment between $\mathbf{p}_{1}$ and $\mathbf{p}_{3} .$

a. Trueb. falsec. True

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 6

Curves and Surfaces

Vectors

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forgiven a statement and were asked to market tour false and then to just fire answer in part a. The statement is Cube in. Our three has exactly five facets. The statement is false. A number of facets of a cube in our three is six because in our three, a facet is the same as a square face. Now, in part B, the statement is a point. P is an extreme point of a poly tope if and only if p is a vertex of the politics. This statement is true. This is a direct consequence of the're um 14. In the book, the serum says that a point of a poly tope is an extreme point if and only if it's a vertex and definitely if it lies in the profile now in part C were given the statement if s is a non empty, compact convex set and a linear functional attains its maximum at a point p then P is an extreme point of s. This is false. So while the maximum is always attained at some extreme point maybe other points that are not extreme points I wish the maximum is attained. In fact, there are examples given in the book. I would say, for example, look, at example three part C. After the're, um 16. We see that the linear functional attendance maximum value at to extreme points, but then also at every point in the complex whole of these extreme points and these points in the comics whole are not. Extreme points now in Part D were given the statement a two dimensional Polito always has the same number of Vergis is and edges. This statement is true to see why. Consider Oilers formula so by Oilers formula checking in to be to We get that for a two dimensional polito f zero of call this P and then minus If one of P is equal to one plus negative one to the first, which is zero So it follows F zero p is equal to F one of p. Therefore, the number of Vergis is for this play. Tope is always the same as the number of edges

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