For each of the following sets, prove whether they are convex or not. 1. Given a collection of n = N convex sets {S}"ï±1, S := 2. Given p > 0, S := {x = Rd : ||x||₂ = 1}. € i=1' ±1 Si. i=1 3. Given p≥ 1, S := {{a¡};£1, a¡ € R : (Σºº1₁ |ai|P)½³ < ∞0}.

Let ƒ be defined on an interval (a, b) and suppose that ƒ(c) ≠ 0 at some c where ƒ is continuous. Show that there is an interval (c - d, c + d) about c where ƒ has the same sign as ƒ(c)..

Find the length of the curve over the given interval. r = 8, [0, π/6]

Find the open intervals on which r(t) = t3i +t4j is smooth.

FIND THE A BAR, ADHERENT POINTS OF A,  CLUSTER POINT OF A, INT(A), EXT(A), FR(A), DENSE OR NOT?

Suppose f is differentiable on an interval containing a, b, and let P(a,f(a)) and Q(b,f(b)) be distinct points on the graph of f.  Let c be the x-coordinate of teh point at which the lines tangent to the curve of P and Q intersect, assuming that the tangent lines are not parallel (see figure). Complete the following: a. if f(x) = x^2, show that c = (a + b)/2, the arithemtic mean of a and b, for real numbers a and b. The variable c is defined as the x-value of the point of intersection of the lines tangent to the points P and Q. Compute the slope of line tangent to point P. What is the equation of the line tangent to point P? Compute the slope of the line tangent to point Q. What is the equation of the line tangent to the point Q? Find the point of intersection of the two lines. b. if f(x)=sqrt(x), show that c = sqrt(ab), the geometric mean of a and b, for a > 0 and b > 0. Compute the derivative of f(x) Compute the slope of the line tangent to point P. What is the equation of the…

In the adjacent figure, ΑΒΓΔ is a trapezoid (AB II ΓΔ) and EZ is its median. The non-parallel sides of AΔ and ΒΓ intersect at K. H, Θ are the means of ΚΑ and ΚΒ respectively. Prove that EZΗΘ is a trapezoid.

Verify that the Intermediate Value Theorem applies to the indicated interval and find the calue of c guaranteed by the therom. f(x)=(x-6)3+4, [4,7], f(c)=3

Is the following bounded or unbounded and why? f(x,y) = ln(x2 + y2)

Consider the following functions defined on (−π/2,π/2). Match the convexity. If the function is neither convex nor concave over the interval, select NA   |x|x   x−|x|    |x|+x   sinx   cosx