Let h(x)=x*sin(x). Below is Eliza's attempt to write a formal justification for the fact that the equation h(x)=1 has a solution where 0

Question

Let  h(x)=x*sin(x). Below is Eliza's attempt to write a formal justification for the fact that the equation  h(x)=1 has a solution where  0 <= x <= 2. Is Eliza's justification complete? If not, why? Eliza's justification:  h is defined for all real numbers, and polynomial and trigonometric functions are continuous at all points in their domains. Furthermore,  h(0)=0 and  h(2)~~1.82, so 1 is between  h(0) and  h(2). So, according to the intermediate value theorem,  h(x)=1 must have a solution somewhere between  x=0 and  x=2. Choose 1 answer: (A) Yes, Eliza's justification is complete. (B) No, Eliza didn't establish that 1 is between  h(0) and  h(2). (C) No, Eliza didn't establish that  h is continuous.

Bytelearn · Accepted Answer

Function Definition: Eliza's first claim is that the function h(x) = x*sin(x) is defined for all real numbers. This is true because both the polynomial part (x) and the trigonometric part (sin(x)) are defined for all real numbers. Continuity of Functions: Eliza's second claim is that polynomial and trigonometric functions are continuous at all points in their domains. This is also true; the product of two continuous functions is continuous, so h(x) is continuous everywhere. Value at x=0: Eliza's third claim is that h(0) = 0. This is correct because sin(0) = 0, and thus 0 * sin(0) = 0. Value at x=2: Eliza's fourth claim is that h(2) is approximately 1.82. To verify this, we need to calculate h(2) = 2*sin(2). Since sin(2) is not a well-known exact value, we can use a calculator or approximation to find that sin(2) is approximately 0.9093. Therefore, h(2) ≈ 2 * 0.9093 = 1.8186, which rounds to approximately 1.82. Intermediate Value Theorem: Eliza's fifth claim is that since 1 is between h(0) and h(2), the intermediate value theorem guarantees that there is at least one solution to h(x) = 1 for some x in the interval [0, 2]. This is true because the intermediate value theorem states that if a function is continuous on a closed interval [a, b] and N is any number between f(a) and f(b), then there is at least one number c in the interval [a, b] such that f(c) = N. Justification Complete: Eliza's justification seems complete because she has established that h is continuous, h(0) = 0, h(2) is approximately 1.82, and 1 is between h(0) and h(2). Therefore, by the intermediate value theorem, there must be a solution to h(x) = 1 in the interval [0, 2].

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