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The global maxima of f(x) = [2{-x2 + x + 1}] is , where {x} denotes fractional part of x and [-] denotes greatest integer function
- a)2
- b)1
- c)0
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
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The global maxima of f(x) = [2{-x2 + x + 1}] is , where {x} denotes fr...
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global maxima of f(x) is 1.
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The global maxima of f(x) = [2{-x2 + x + 1}] is , where {x} denotes fr...
Explanation:
To find the global maxima of the function f(x) = 2{-x^2 + x + 1}, we need to determine the values of x that yield the highest possible output for the function.
The function f(x) involves the fractional part {x}, which means that it takes the decimal part of any real number x. The fractional part can be defined as x - ⌊x⌋, where ⌊x⌋ represents the greatest integer function.
Finding the Possible Range of x:
To determine the possible range of x, we need to consider the domain of the function. Since the function involves the fractional part, we know that the fractional part of any real number x is always between 0 and 1. Therefore, the possible range of x is [0, 1].
Finding the Critical Points:
To find the critical points of the function, we need to find the values of x where the derivative of the function is equal to zero or does not exist.
Taking the derivative of f(x) with respect to x, we get:
f'(x) = 2(-2x + 1)
Setting f'(x) equal to zero and solving for x, we get:
-2x + 1 = 0
-2x = -1
x = 1/2
Evaluating the Function at Critical Points and Endpoints:
Now, we need to evaluate the function at the critical point x = 1/2 and the endpoints of the possible range, x = 0 and x = 1.
f(0) = 2{-0^2 + 0 + 1} = 2{1} = 0
f(1/2) = 2{-1/4 + 1/2 + 1} = 2{1/4} = 1/2
f(1) = 2{-1^2 + 1 + 1} = 2{1} = 0
Determining the Global Maxima:
From the evaluations, we can see that the function f(x) achieves its highest value of 1/2 at x = 1/2. Therefore, the global maxima of the function is 1/2.
Hence, the correct answer is option B: 1.
To find the global maxima of the function f(x) = 2{-x^2 + x + 1}, we need to determine the values of x that yield the highest possible output for the function.
The function f(x) involves the fractional part {x}, which means that it takes the decimal part of any real number x. The fractional part can be defined as x - ⌊x⌋, where ⌊x⌋ represents the greatest integer function.
Finding the Possible Range of x:
To determine the possible range of x, we need to consider the domain of the function. Since the function involves the fractional part, we know that the fractional part of any real number x is always between 0 and 1. Therefore, the possible range of x is [0, 1].
Finding the Critical Points:
To find the critical points of the function, we need to find the values of x where the derivative of the function is equal to zero or does not exist.
Taking the derivative of f(x) with respect to x, we get:
f'(x) = 2(-2x + 1)
Setting f'(x) equal to zero and solving for x, we get:
-2x + 1 = 0
-2x = -1
x = 1/2
Evaluating the Function at Critical Points and Endpoints:
Now, we need to evaluate the function at the critical point x = 1/2 and the endpoints of the possible range, x = 0 and x = 1.
f(0) = 2{-0^2 + 0 + 1} = 2{1} = 0
f(1/2) = 2{-1/4 + 1/2 + 1} = 2{1/4} = 1/2
f(1) = 2{-1^2 + 1 + 1} = 2{1} = 0
Determining the Global Maxima:
From the evaluations, we can see that the function f(x) achieves its highest value of 1/2 at x = 1/2. Therefore, the global maxima of the function is 1/2.
Hence, the correct answer is option B: 1.
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