Let f: [0, ∞0)→ [0, ∞) be the square-root function, that is, f(x)=√x for every nonnegative real number . Evaluate the following proposed proof that f is not differentiable at 0. 1. Seeking a contradiction, suppose that f'(0) does exist. 2. Let g: [0, ∞0)→ [0, ∞o) be the function defined by g(x) = f(x) f(x) for every nonnegative real number . 3. By the product rule, g/ (0) exists and equals 2 f(0) ƒ' (0), that is, g (0) = 0. 4. Step (2) implies that g(x) = x for every nonnegative real number *, so g(x) = 1 for every , and in particular g/ (0) = 1. 5. The conclusions of steps (3) and (4) are incompatible, and the contradiction means that f'(0) cannot exist after all. Which of the following statements best describes this proposed proof? Step 2 is faulty. The proof is valid. Step 3 is faulty. Step 4 is faulty. Step 1 is faulty. Step 5 is faulty.
Let f: [0, ∞0)→ [0, ∞) be the square-root function, that is, f(x)=√x for every nonnegative real number . Evaluate the following proposed proof that f is not differentiable at 0. 1. Seeking a contradiction, suppose that f'(0) does exist. 2. Let g: [0, ∞0)→ [0, ∞o) be the function defined by g(x) = f(x) f(x) for every nonnegative real number . 3. By the product rule, g/ (0) exists and equals 2 f(0) ƒ' (0), that is, g (0) = 0. 4. Step (2) implies that g(x) = x for every nonnegative real number *, so g(x) = 1 for every , and in particular g/ (0) = 1. 5. The conclusions of steps (3) and (4) are incompatible, and the contradiction means that f'(0) cannot exist after all. Which of the following statements best describes this proposed proof? Step 2 is faulty. The proof is valid. Step 3 is faulty. Step 4 is faulty. Step 1 is faulty. Step 5 is faulty.
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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Transcribed Image Text:Let f: [0, ∞0)→ [0, ∞) be the square-root function, that is, f(x)=√x
for every nonnegative real number . Evaluate the following proposed proof
that f is not differentiable at 0.
1. Seeking a contradiction, suppose that f'(0) does exist.
2. Let g: [0, ∞) → [0, ∞0) be the function defined by
g(x) = f(x) f(x) for every nonnegative real number .
3. By the product rule, g/(0) exists and equals 2 f(0) f'(0), that is,
g (0) = 0.
4. Step (2) implies that g(x) = x for every nonnegative real number,
so g(x) = 1 for every , and in particular g/ (0) = 1.
5. The conclusions of steps (3) and (4) are incompatible, and the
contradiction means that f'(0) cannot exist after all.
Which of the following statements best describes this proposed proof?
Step 2 is faulty.
The proof is valid.
Step 3 is faulty.
Step 4 is faulty.
Step 1 is faulty.
Step 5 is faulty.
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