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Let 
h(x)=log_(2)(x)*x^(2).
Below is Jasmine's attempt to write a formal justification for the fact that the equation 
h(x)=3 has a solution where 
1 <= x <= 2.
Is Jasmine's justification complete?
If not, why?
Jasmine's justification:

h(1)=0 and

h(2)=4, so 3 is between 
h(1) and 
h(2).
So, according to the intermediate value theorem, 
h(x)=3 must have a solution for an 
x-value within the interval 
[1,2].
Choose 1 answer:
(A) Yes, Jasmine's justification is complete.
(B) No, Jasmine didn't establish that 3 is between 
h(1) and 
h(2).
(C) No, Jasmine didn't establish that 
h is continuous.

Let h(x)=log2(x)x2 h(x)=\log _{2}(x) \cdot x^{2} .\newlineBelow is Jasmine's attempt to write a formal justification for the fact that the equation h(x)=3 h(x)=3 has a solution where 1x2 1 \leq x \leq 2 .\newlineIs Jasmine's justification complete?\newlineIf not, why?\newlineJasmine's justification:\newlineh(1)=0 h(1)=0 and\newlineh(2)=4 h(2)=4 , so 33 is between h(1) h(1) and h(2) h(2) .\newlineSo, according to the intermediate value theorem, h(x)=3 h(x)=3 must have a solution for an x x -value within the interval [1,2] [1,2] .\newlineChoose 11 answer:\newline(A) Yes, Jasmine's justification is complete.\newline(B) No, Jasmine didn't establish that 33 is between h(1) h(1) and h(2) h(2) .\newline(C) No, Jasmine didn't establish that h(x)=3 h(x)=3 22 is continuous.

Full solution

Q. Let h(x)=log2(x)x2 h(x)=\log _{2}(x) \cdot x^{2} .\newlineBelow is Jasmine's attempt to write a formal justification for the fact that the equation h(x)=3 h(x)=3 has a solution where 1x2 1 \leq x \leq 2 .\newlineIs Jasmine's justification complete?\newlineIf not, why?\newlineJasmine's justification:\newlineh(1)=0 h(1)=0 and\newlineh(2)=4 h(2)=4 , so 33 is between h(1) h(1) and h(2) h(2) .\newlineSo, according to the intermediate value theorem, h(x)=3 h(x)=3 must have a solution for an x x -value within the interval [1,2] [1,2] .\newlineChoose 11 answer:\newline(A) Yes, Jasmine's justification is complete.\newline(B) No, Jasmine didn't establish that 33 is between h(1) h(1) and h(2) h(2) .\newline(C) No, Jasmine didn't establish that h(x)=3 h(x)=3 22 is continuous.
  1. Understand Function and IVT: Understand the function and the Intermediate Value Theorem (IVT). The function h(x)=log2(x)x2h(x) = \log_2(x) \cdot x^2 is a product of a logarithmic function and a power function. The IVT states that if a function is continuous on a closed interval [a,b][a, b] and kk is any number between f(a)f(a) and f(b)f(b), then there is at least one number cc in the interval [a,b][a, b] such that f(c)=kf(c) = k.
  2. Check Endpoint Values: Check the values of h(x)h(x) at the endpoints of the interval.\newlineJasmine calculated h(1)h(1) and h(2)h(2). We need to verify these calculations.\newlineh(1)=log2(1)12=01=0h(1) = \log_2(1) \cdot 1^2 = 0 \cdot 1 = 0\newlineh(2)=log2(2)22=14=4h(2) = \log_2(2) \cdot 2^2 = 1 \cdot 4 = 4\newlineThese calculations are correct, and they show that h(1)=0h(1) = 0 and h(2)=4h(2) = 4.
  3. Check Value 33: Determine if the value 33 is between h(1)h(1) and h(2)h(2).\newlineSince h(1)=0h(1) = 0 and h(2)=4h(2) = 4, the value 33 is indeed between h(1)h(1) and h(2)h(2). This part of Jasmine's justification is correct.
  4. Verify Continuity: Verify the continuity of the function h(x)h(x) on the interval [1,2][1, 2]. Jasmine's justification assumes that h(x)h(x) is continuous on the interval [1,2][1, 2] but does not explicitly state it. The function h(x)h(x) is continuous on the interval [1,2][1, 2] because both the logarithmic function log2(x)\log_2(x) and the power function x2x^2 are continuous on that interval, and the product of continuous functions is also continuous. However, Jasmine did not explicitly mention this in her justification.

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