Suppose f: (0, ∞) → (0, ∞) and g: (0, ∞) → (0, ∞) andh: (0, ∞) → (0, ∞) are continuous functions such that h(x) = f(x)g(x) forevery positive real number .True or false: If f and h are differentiable, then g must be differentiable too.TrueFalse

Given f(x) and h(x) are differentiable for all positive real number. And h(x) = f(x)g(x) for all…

Answered: Suppose f: (0, ∞) → (0, ∞) and g: (0,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(d) Determine the values of x on (−4, 4) for which f(x) is not continuous.(e) Determine the values of x on (−4, 4) for which f(x) is not differentiable.(f) Determine the values of x for which f has a horizontal tangent. image attached
Find a and b so that f(x) is continuous if f (x) = Ⓒa = 1/2b = 60 96 a=- - 21/1/ b = 2/20 Ⓒa = -¹/ b = ²/7/20 a = 96 b = -4/12 2x² + 3x ax+b -x³ + 4x² x <-4 -4
Let f be a real-valued continuous and differentiable function. Let function g be defined by g(x) = f(|x| + 2). A student presents the following proof to show that there exists a real number c € (-1, 1) such that g/(c) = 0. (1) Since f is a continuous function, so is g over the interval [-1, 1]. (II) Since f is differentiable, so is g over the interval (-1, 1). (III) It is evident from the definition of g that g(−1) = g(1). (IV) If the above conditions hold, then by Rolle's theorem, there exists dg(x) c = (-1, 1) such that gl (c) dx |x=c Which statement about this proof is correct? = 0. Step (1) does not hold, and hence Rolle's theorem does not apply. Step (II) does not hold, and hence Rolle's theorem does not apply. Step (III) does not hold, and hence Rolle's theorem does not apply. Step (IV) does not hold, and hence the conclusion is false. The proof is completely correct, and the conclusion holds.
1 (d)
2- Determine the value of m such that the function x - y - 5 yE + 5 y ; x + y f(r, y) = is continuous at (0,0). m ;x = y
Theorem: Let f,g e C(X)are continuous function then 4) tf is continuous function and (tf)(x) = tf(x) 5) \f| is continuous function and |f\ (x) = |f(x)|
Suppose f: (0, ∞) → (0, ∞) is a continuous function, and g: (0, ∞) → (0,00) is defined by the property that g(x) = (f(x))² for every positive real number . True or false: If g is differentiable, then f must be differentiable too. True False

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Suppose f: (0, ∞) → (0, ∞) and g: (0, ∞) → (0, ∞) and h: (0, ∞) → (0, ∞) are continuous functions such that h(x) = f(x)g(x) for every positive real number . True or false: If f and h are differentiable, then g must be differentiable too. True False