1. Consider the paraboloid f(r, v) = az + bry + cy", where we will assume that a, e 0. We will investigate the behavior of f at its critical point. (a) Show that (0, 0) is the only critical point when - dac i 0. (b) By completing the square, show that 4ac - S(a, M) =a (c) Suppose that D= dac - 6. Without using the second derivative test: i. Suppose that D >0 and a > 0. Show that / has a local minimum at (0,0). [Hint: Show that S(0,0) = 0. Use the fact that a and D are both positive to conclude that when z, y 0, /(2, v) > 0. i. Suppose that D>0 and a <0. Show that / has a local maximum at (0,0). ii. Finally, suppose that D<0. Show that / has a saddle point. [Hint: Explain why the tangent plane at (0,0) has equation z= 0. We wish to show that / crosses this tangent plane, by showing there exist different paths for which / has opposite signs along those paths.) (d) Point out that the value D as described above is exactly the function D(a, v) involved in the second derivative test, and that the results match that test.
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
I just need question 1, c and d
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