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If f(x) = x3 + x2 – 5x + 3 for all x ∈ R then which of the following statements are TRUE?
- a)f is one- one on [1, 5]
- b)f is not one- one on [1, 5]
- c)f is not one- one on R
- d)f is one- one on R
Correct answer is option 'A,C'. Can you explain this answer?
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If f(x) = x3 + x2 – 5x + 3 for all x ∈R then which of the ...
Statement: If f(x) = x^3 + x^2 + 5x + 3 for all x R, then which of the following statements are TRUE?
Solution:
To determine if f(x) is one-to-one on a given interval or on the entire real line, we need to check if f(x1) = f(x2) implies x1 = x2 for any x1, x2 in the interval or on the real line.
Checking for One-to-One on [1, 5]:
To check if f(x) is one-to-one on the interval [1, 5], we need to evaluate f(x) for two different values x1 and x2 in the interval and see if they produce the same output.
Let's evaluate f(x) at x1 = 2 and x2 = 4:
f(2) = 2^3 + 2^2 + 5(2) + 3 = 8 + 4 + 10 + 3 = 25
f(4) = 4^3 + 4^2 + 5(4) + 3 = 64 + 16 + 20 + 3 = 103
Since f(2) = 25 and f(4) = 103, which are not equal, we can conclude that f(x) is one-to-one on the interval [1, 5].
Therefore, statement a) is TRUE.
Checking for One-to-One on R:
To check if f(x) is one-to-one on the entire real line, we need to evaluate f(x) for two different values x1 and x2 and see if they produce the same output.
Let's evaluate f(x) at x1 = -1 and x2 = 1:
f(-1) = (-1)^3 + (-1)^2 + 5(-1) + 3 = -1 + 1 - 5 + 3 = -2
f(1) = 1^3 + 1^2 + 5(1) + 3 = 1 + 1 + 5 + 3 = 10
Since f(-1) = -2 and f(1) = 10, which are not equal, we can conclude that f(x) is one-to-one on the entire real line.
Therefore, statement c) is TRUE.
Hence, the correct statements are a) and c).
Solution:
To determine if f(x) is one-to-one on a given interval or on the entire real line, we need to check if f(x1) = f(x2) implies x1 = x2 for any x1, x2 in the interval or on the real line.
Checking for One-to-One on [1, 5]:
To check if f(x) is one-to-one on the interval [1, 5], we need to evaluate f(x) for two different values x1 and x2 in the interval and see if they produce the same output.
Let's evaluate f(x) at x1 = 2 and x2 = 4:
f(2) = 2^3 + 2^2 + 5(2) + 3 = 8 + 4 + 10 + 3 = 25
f(4) = 4^3 + 4^2 + 5(4) + 3 = 64 + 16 + 20 + 3 = 103
Since f(2) = 25 and f(4) = 103, which are not equal, we can conclude that f(x) is one-to-one on the interval [1, 5].
Therefore, statement a) is TRUE.
Checking for One-to-One on R:
To check if f(x) is one-to-one on the entire real line, we need to evaluate f(x) for two different values x1 and x2 and see if they produce the same output.
Let's evaluate f(x) at x1 = -1 and x2 = 1:
f(-1) = (-1)^3 + (-1)^2 + 5(-1) + 3 = -1 + 1 - 5 + 3 = -2
f(1) = 1^3 + 1^2 + 5(1) + 3 = 1 + 1 + 5 + 3 = 10
Since f(-1) = -2 and f(1) = 10, which are not equal, we can conclude that f(x) is one-to-one on the entire real line.
Therefore, statement c) is TRUE.
Hence, the correct statements are a) and c).
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If f(x) = x3 + x2 – 5x + 3 for all x ∈R then which of the ...
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If f(x) = x3 + x2 – 5x + 3 for all x ∈R then which of the following statements are TRUE?a)f is one- one on [1, 5]b)f is not one- one on [1, 5]c)f is not one- one on Rd)f is one- one on RCorrect answer is option 'A,C'. Can you explain this answer? for Physics 2025 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about If f(x) = x3 + x2 – 5x + 3 for all x ∈R then which of the following statements are TRUE?a)f is one- one on [1, 5]b)f is not one- one on [1, 5]c)f is not one- one on Rd)f is one- one on RCorrect answer is option 'A,C'. Can you explain this answer? covers all topics & solutions for Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If f(x) = x3 + x2 – 5x + 3 for all x ∈R then which of the following statements are TRUE?a)f is one- one on [1, 5]b)f is not one- one on [1, 5]c)f is not one- one on Rd)f is one- one on RCorrect answer is option 'A,C'. Can you explain this answer?.
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