Prove that the function f(x) = (1 + x)3/2 – 34 – 1 is increasing on...
6. (25) Is the function f(x) below integrable on the interval (0, 3)? Prove your answer using upper sums and lower sums, and if f is integrable, find Sof by computing L(f) or U(f) directly. f(x) = 0 <0 x +1 0 < x < 1 x=1 2 > 1 I 1 Ž
Let f(x) = x^(1/3) with domain (0,infinity). Prove, by epsilon-delta language, that f is continuous at c in an element of (0, infinity). 2. Let f(0) = 25 with domain (0,00). Prove, by the e-8 language, that f is continuous at CE (0,0)
Q3 (Prove that P∞ k=1 1/kr < ∞ if r > 1) . Let f : (0,∞) → R be a twice differentiable function with f ''(x) ≥ 0 for all x ∈ (0,∞). (a) Show that f '(k) ≤ f(k + 1) − f(k) ≤ f '(k + 1) for all k ∈ N. (b) Use (a), show that Xn−1 k=1 f '(k) ≤ f(n) − f(1) ≤ Xn k=2 f '(k). (c) Let r > 1. By finding...
Let f : [0,∞) → R be the function defined by f ( x ) = 2 ⌊ x ⌋ − x? where x? = x − ⌊x⌋ is the decimal part of x. Prove that f is injective. Let f: 0,00) + R be the function defined by f(3) = 212) where ã = x — [x] is the decimal part of x. Prove that f is injective.
A function f:R HR is said to be strictly increasing if f(x1) < f(12) whenever I] < 12. Prove: If a differentiable function f is strictly increasing, then f'(x) > 0. Then give counterexamples to show that the following statements are false, in general. (i) If a differentiable function f is strictly increasing, then f'(2) >0 for all 1. (ii) If f'(x) > 0 for all x, then f is strictly increasing -
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
Consider the function f(x) = 14x2 + 200 on the open interval (0,00). (1) Find the critical value(s) off on the open interval (0, 0). If more than one, then list them separated by commas. Critical value(s) = Preview (2) Find f''(x) = Preview (3) Looking at f''(x) we can conclude the following: f''(x) > 0 for all 3 on the interval (0,0) and thus we have an absolute maximum at the critical value f''(x) < 0 for all x...
1a) A production function has the form f(a,b) = a^2 x b^3 . Does this function exhibit constant, increasing, or decreasing returns to scale? 1b)A production function has the form f(a,b) = 3a^1/2 x b^1/2. Does this function exhibit constant, increasing, or decreasing returns to scale? Explain. Thank you.
7. Consider the function f:R + R defined by f(x) = x < 0, 3 > 0. e-1/x2, Prove that f is differentiable of all orders and that f(n)(0) = 0 for all n e N. Conclude that f does not have a convergent power series expansion En Anx" for x near the origin. [We will see later in this class that this is impossible for holomorphic functions, namely being (complex) differentiable implies that there is always a convergent power...
Let f(3) = 1 (a) Prove {f} 1 + nx converges to 0 pointwise on (-0,00). (b) Prove or disprove {n} , converges to 0 uniformly on (-0, 0);