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- Minimum sum Find positive numbers x and y satisfying the equa-tion xy = 12 such that the sum 2x + y is as small as possible.arrow_forward(Lipschitz Functions). A function f : A → R is called Lipschitz if there exists a bound M >0 such that |( f(x) − f(y)/x − y)| ≤ M for all x not equal to y ∈ A. Geometrically speaking, a function f is Lipschitz if there is a uniform bound on the magnitude of the slopes of lines drawn through any two points on the graph of f. (a) Show that if f : A → R is Lipschitz, then it is uniformly continuous on A. (b) Is the converse statement true? Are all uniformly continuous functionsnecessarily Lipschitz?arrow_forwardThe rectangles in the graph below illustrate a left endpoint Riemann sum for f(x)=(x^2)/(12) f(x)=(x^2)/(12) on the interval [2,6].The value of this left endpoint Riemann sum is __________________ , the area of the region enclosed by y=f(x)y=f(x), the x-axis, and the vertical lines x = 2 and x = 6.arrow_forward
- Real Analysis Is the proof for this claim correct? State that it is correct, or circle the first error you see if it isn't, and explain if it can be corrected to be proven true. Claim: If f: (0,1] → R and g: [1,2) → R are uniformly continuous on their domains, and f(1) =g(1), then the function h: (0,2) → R, defined by h(x) =f(x) for x ∈ (0,1] and h(x)=g(x) for x ∈ [1,2), is uniformly continuous on (0,2). Proof: Let ε >0. Since f is uniformly continuous on (0,1], there exists δ1 > 0 such that if x,y ∈ (0,1] and |x−y|< δ1, then |f(x)−f(y)|< ε/2. Since g is uniformly continuous on [1,2), there exists δ2 > 0 such that if x,y ∈ [1,2) and |x−y| < δ2, then |g(x)−g(y)| < ε/2. Let δ= min{δ1, δ2}. Now suppose x,y ∈ (0,2) with x < y and |x−y|< δ. If x,y ∈ (0,1], then |x−y| < δ ≤ δ1 and so |h(x)−h(y)| = |f(x)−f(y)| < ε/2 < ε. If x,y ∈ [1,2), then |x−y| < δ ≤ δ2 and so |h(x)−h(y)| = |g(x)−g(y)| < ε/2 < ε. If x ∈ (0,1) and y ∈ (1,2), then |x−1| <…arrow_forwardreal analysis show all work/steps please. Suppose f is uniformly continuous on [n, n+ 1] ∀ n ∈ Z. Does it follow that f is uniformly continuous on R?arrow_forwardFermat’s Theorem: Let f be a continuous function on an open interval I where c is in I . If f has an extremum at c, then c must be a critical number of f. Expound this theorem.arrow_forward
- Evaluate the Riemann sum for f(x) = x3 − 6x + 2, taking the sample points to be midpoint and a = 0, b = 4, and n = 8b) Find the exact value of ∫03 f(x) dx using Riemann sum c) Find the exact value of ∫03 f(x) dx using the Fundamental Theorem of Calculusarrow_forward(Term-by-term Differentiability Theorem). Let fn be differentiable functions defined on an interval A, and assume ∞ n=1 fn(x) converges uniformly to a limit g(x) on A. If there exists a point x0 ∈ [a, b] where ∞ n=1 fn(x0) converges, then the series ∞ n=1 fn(x) converges uniformly to a differentiable function f(x) satisfying f(x) = g(x) on A. In other words, Proof. Apply the stronger form of the Differentiable Limit Theorem (Theorem6.3.3) to the partial sums sk = f1 + f2 + · · · + fk. Observe that Theorem 5.2.4 implies that sk = f1 + f2 + · · · + fk . In the vocabulary of infinite series, the Cauchy Criterion takes the followingform.arrow_forwarda) Give an example of a function f : [−2, 3] → R which is not continuous at 1 but which is integrable b) Give an example of a function f : [−2, 2] → R which is not differentiable at −1 but which is continuous at −1 - please include all steps and working with explanationarrow_forward
- Real Analysis T/F questions, no need to prove 1) Every bounded continuous function attains its maximum and minimum values. 2) Every infinite set is unbounded. 3) If A is a set of real numbers that is not open, then A is closed. 4) If f is continuous on [a,b], then f is uniformly continuous on (a,b).arrow_forwarda) Use the limit process to find the area of the region bounded by the graph of the function and the x-axis over the given interval; b) Use the fundamental theorem of calculus to verify your result c) Find the average value of the function over the given interval. f(x)=3x2+2x+1, [1,4] I'd like to know how to do part Barrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageCollege AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningFunctions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning