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f(x) = max(2 - 5x, 3x + 4), |y - 1| < 4a, where a is the minimum value of f(x). How many possible values of y are perfect squares greater than 1?
Correct answer is '2'. Can you explain this answer?
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f(x) = max(2 - 5x, 3x + 4), |y - 1| 4a, where a is the minimum value o...
f(x) = max(2 - 5x, 3x + 4)
The minimum value of f(x) is obtained when 2 - 5x = 3x + 4
That is when x = -1/4
4a = 4f(-1/4) = 4x13/4 = 13
So | y - 1| < 13
-13 < y - 1 <13
-12 < y < 14
The perfect square values of y (greater than 1) in the given range are 4, and 9.
There are 2 such perfect square values of y.
The minimum value of f(x) is obtained when 2 - 5x = 3x + 4
That is when x = -1/4
4a = 4f(-1/4) = 4x13/4 = 13
So | y - 1| < 13
-13 < y - 1 <13
-12 < y < 14
The perfect square values of y (greater than 1) in the given range are 4, and 9.
There are 2 such perfect square values of y.
Answer: 2
Most Upvoted Answer
f(x) = max(2 - 5x, 3x + 4), |y - 1| 4a, where a is the minimum value o...
To find the number of possible values of y that are perfect squares greater than 1, we need to analyze the given function f(x) = max(2 - 5x, 3x - 4) and the inequality |y - 1| ≤ 4a.
Determining the Minimum Value of f(x)
To find the minimum value of f(x), we need to determine the critical points of the function. The critical points occur when the two expressions inside the max function are equal.
2 - 5x = 3x - 4
Simplifying the equation, we get:
8x = 6
x = 3/4
Substituting this value of x back into the function, we get:
f(3/4) = max(2 - 5(3/4), 3(3/4) - 4)
= max(-13/4, -1/4)
= -1/4
Therefore, the minimum value of f(x) is -1/4.
Analyzing the Inequality |y - 1| ≤ 4a
The inequality |y - 1| ≤ 4a can be rewritten as -4a ≤ y - 1 ≤ 4a.
Since a is the minimum value of f(x) and we have already determined that the minimum value of f(x) is -1/4, we can substitute -1/4 for a in the inequality:
-4(-1/4) ≤ y - 1 ≤ 4(-1/4)
1 ≤ y - 1 ≤ -1
Simplifying the inequality, we get:
2 ≤ y ≤ 0
Finding Perfect Square Values of y
To find the perfect square values of y that are greater than 1, we need to consider the range of y values.
Since 2 ≤ y ≤ 0 is not possible, there are no perfect square values of y that satisfy the given conditions.
Conclusion
Hence, based on the given function f(x) = max(2 - 5x, 3x - 4) and the inequality |y - 1| ≤ 4a, there are no possible values of y that are perfect squares greater than 1. Therefore, the correct answer is 0, not 2.
Determining the Minimum Value of f(x)
To find the minimum value of f(x), we need to determine the critical points of the function. The critical points occur when the two expressions inside the max function are equal.
2 - 5x = 3x - 4
Simplifying the equation, we get:
8x = 6
x = 3/4
Substituting this value of x back into the function, we get:
f(3/4) = max(2 - 5(3/4), 3(3/4) - 4)
= max(-13/4, -1/4)
= -1/4
Therefore, the minimum value of f(x) is -1/4.
Analyzing the Inequality |y - 1| ≤ 4a
The inequality |y - 1| ≤ 4a can be rewritten as -4a ≤ y - 1 ≤ 4a.
Since a is the minimum value of f(x) and we have already determined that the minimum value of f(x) is -1/4, we can substitute -1/4 for a in the inequality:
-4(-1/4) ≤ y - 1 ≤ 4(-1/4)
1 ≤ y - 1 ≤ -1
Simplifying the inequality, we get:
2 ≤ y ≤ 0
Finding Perfect Square Values of y
To find the perfect square values of y that are greater than 1, we need to consider the range of y values.
Since 2 ≤ y ≤ 0 is not possible, there are no perfect square values of y that satisfy the given conditions.
Conclusion
Hence, based on the given function f(x) = max(2 - 5x, 3x - 4) and the inequality |y - 1| ≤ 4a, there are no possible values of y that are perfect squares greater than 1. Therefore, the correct answer is 0, not 2.
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f(x) = max(2 - 5x, 3x + 4), |y - 1| 4a, where a is the minimum value of f(x). How many possible values of y are perfect squares greater than 1?Correct answer is '2'. Can you explain this answer?
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f(x) = max(2 - 5x, 3x + 4), |y - 1| 4a, where a is the minimum value of f(x). How many possible values of y are perfect squares greater than 1?Correct answer is '2'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about f(x) = max(2 - 5x, 3x + 4), |y - 1| 4a, where a is the minimum value of f(x). How many possible values of y are perfect squares greater than 1?Correct answer is '2'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for f(x) = max(2 - 5x, 3x + 4), |y - 1| 4a, where a is the minimum value of f(x). How many possible values of y are perfect squares greater than 1?Correct answer is '2'. Can you explain this answer?.
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