Let f: [0 1] → R be a function of the class c ^ 2 that...

Let f: [0 1] → R be a function of the class c ^ 2 that satisfies the differential equation f '' (x) = e^xf(x) for all x in (0,1). Show that if x0 is in (0,1) then f can not have a positive local maximum at x0 and can not have a negative local minimum at x0. If f (0) = f (1) = 0, prove that f = 0

Expert Solution

Colby Messinger answered 2 years ago

Let A ⊆ R, let f : A → R be a function, and let c...

Let A ⊆ R, let f : A → R be a function, and let c be a limit point of A. Suppose that a student copied down the following definition of the limit of f at c: “we say that limx→c f(x) = L provided that, for all ε > 0, there exists a δ ≥ 0 such that if 0 < |x − c| < δ and x ∈ A, then |f(x) − L| < ε”. What was...

Let f(x, y) be a function such that f(0, 0) = 1, f(0, 1) = 2,...

Let f(x, y) be a function such that f(0, 0) = 1, f(0, 1) = 2, f(1, 0) = 3, f(1, 1) = 5, f(2, 0) = 5, f(2, 1) = 10. Determine the Lagrange interpolation F(x, y) that interpolates the above data. Use Lagrangian bi-variate interpolation to solve this and also show the working steps.

Suppose a function f : R → R is continuous with f(0) = 1. Show that...

Suppose a function f : R → R is continuous with f(0) = 1. Show that if there is a positive number x0 for which f(x0) = 0, then there is a smallest positive number p for which f(p) = 0. (Hint: Consider the set {x | x > 0, f(x) = 0}.)

Let f : R → R be a function. (a) Prove that f is continuous on...

Let f : R → R be a function. (a) Prove that f is continuous on R if and only if, for every open set U ⊆ R, the preimage f −1 (U) = {x ∈ R : f(x) ∈ U} is open. (b) Use part (a) to prove that if f is continuous on R, its zero set Z(f) = {x ∈ R : f(x) = 0} is closed.

Fit a quadratic function of the form ?(?)=?0+?1?+?2?2 f ( t ) = c 0 +...

Fit a quadratic function of the form ?(?)=?0+?1?+?2?2 f ( t ) = c 0 + c 1 t + c 2 t 2 to the data points (0,−1) ( 0 , − 1 ) , (1,8) ( 1 , 8 ) , (2,−7) ( 2 , − 7 ) , (3,−6) ( 3 , − 6 ) , using least squares.

Let f : [0, 1] → R and suppose that, for all finite subsets of [0,...

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A probability density function on R is a function f :R -> R satisfying (i) f(x)≥0...

A probability density function on R is a function f :R -> R satisfying (i) f(x)≥0 or all x e R and (ii) \int_(-\infty )^(\infty ) f(x)dx = 1. For which value(s) of k e R is the function f(x)= e^(-x^(2))\root(3)(k^(5)) a probability density function? Explain.

6. (a) let f : R → R be a function defined by f(x) = x...

6. (a) let f : R → R be a function defined by f(x) = x + 4 if x ≤ 1 ax + b if 1 < x ≤ 3 3x x 8 if x > 3 Find the values of a and b that makes f(x) continuous on R. [10 marks] (b) Find the derivative of f(x) = tann 1 1 ∞x 1 + x . [15 marks] (c) Find f 0 (x) using logarithmic differentiation, where f(x)...

2. Let f(x) ≥ 0 on [1, 2] and suppose that f is integrable on [1,...

2. Let f(x) ≥ 0 on [1, 2] and suppose that f is integrable on [1, 2] with R 2 1 f(x)dx = 2 3 . Prove that f(x 2 ) is integrable on [1, √ 2] and √ 2 6 ≤ Z √ 2 1 f(x 2 )dx ≤ 1 3 .

5). Let f : [a,b] to R be bounded and f(x) > a > 0, for...

5). Let f : [a,b] to R be bounded and f(x) > a > 0, for all x in [a,b]. Show that if f is Riemann integrable on [a,b] then 1/f : [a,b] to R, (1/f) (x) = 1/f(x) is also Riemann integrable on [a,b].

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