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f(x) = 1 + 2x2 + 4x4 + 6x6 + ................+ 100x100 is polynomial in a real variable x, then f(x) has
- a)Neither a maximum nor a minimum
- b)Only one maximum
- c)Only one minimum
- d)One maximum and one minimum
Correct answer is option 'C'. Can you explain this answer?
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f(x) = 1 + 2x2+ 4x4+ 6x6+ ................+ 100x100is polynomial in a ...
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f(x) = 1 + 2x2+ 4x4+ 6x6+ ................+ 100x100is polynomial in a ...
**Explanation:**
To determine whether the function f(x) = 1 + 2x^2 + 4x^4 + 6x^6 + ... + 100x^100 has a minimum or maximum, we can analyze its behavior as x varies.
**Properties of Even-Power Polynomials:**
An even-power polynomial is a polynomial in which all of the exponents are even. In this case, the exponents range from 2 to 100, and all are even.
For even-power polynomials, the function is always positive or zero, meaning it cannot have a maximum. However, it can have one minimum.
**Proof:**
Let's consider a general even-power polynomial of degree n:
f(x) = a0 + a2x^2 + a4x^4 + ... + anx^n
To analyze its behavior, we need to look at the sign of each term. Since all the exponents are even, the terms will always be positive or zero.
- For x < 0:="" all="" the="" terms="" remain="" positive,="" so="" f(x)="" /> 0.
- For x > 0: All the terms remain positive, so f(x) > 0.
- At x = 0: f(0) = a0, which is a constant term.
Since the function is always positive or zero and does not have any negative terms, it cannot have a maximum. However, it can have a minimum at x = 0 when f(0) = a0.
**Applying the Proof to the Given Polynomial:**
In the given polynomial f(x) = 1 + 2x^2 + 4x^4 + 6x^6 + ... + 100x^100, all the terms are positive or zero. Therefore, the function cannot have a maximum. However, it can have a minimum at x = 0 when f(0) = 1.
Hence, the correct answer is option C: Only one minimum.
To determine whether the function f(x) = 1 + 2x^2 + 4x^4 + 6x^6 + ... + 100x^100 has a minimum or maximum, we can analyze its behavior as x varies.
**Properties of Even-Power Polynomials:**
An even-power polynomial is a polynomial in which all of the exponents are even. In this case, the exponents range from 2 to 100, and all are even.
For even-power polynomials, the function is always positive or zero, meaning it cannot have a maximum. However, it can have one minimum.
**Proof:**
Let's consider a general even-power polynomial of degree n:
f(x) = a0 + a2x^2 + a4x^4 + ... + anx^n
To analyze its behavior, we need to look at the sign of each term. Since all the exponents are even, the terms will always be positive or zero.
- For x < 0:="" all="" the="" terms="" remain="" positive,="" so="" f(x)="" /> 0.
- For x > 0: All the terms remain positive, so f(x) > 0.
- At x = 0: f(0) = a0, which is a constant term.
Since the function is always positive or zero and does not have any negative terms, it cannot have a maximum. However, it can have a minimum at x = 0 when f(0) = a0.
**Applying the Proof to the Given Polynomial:**
In the given polynomial f(x) = 1 + 2x^2 + 4x^4 + 6x^6 + ... + 100x^100, all the terms are positive or zero. Therefore, the function cannot have a maximum. However, it can have a minimum at x = 0 when f(0) = 1.
Hence, the correct answer is option C: Only one minimum.
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f(x) = 1 + 2x2+ 4x4+ 6x6+ ................+ 100x100is polynomial in a real variable x, then f(x) hasa)Neither a maximum nor a minimumb)Only one maximumc)Only one minimumd)One maximum and one minimumCorrect answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about f(x) = 1 + 2x2+ 4x4+ 6x6+ ................+ 100x100is polynomial in a real variable x, then f(x) hasa)Neither a maximum nor a minimumb)Only one maximumc)Only one minimumd)One maximum and one minimumCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for f(x) = 1 + 2x2+ 4x4+ 6x6+ ................+ 100x100is polynomial in a real variable x, then f(x) hasa)Neither a maximum nor a minimumb)Only one maximumc)Only one minimumd)One maximum and one minimumCorrect answer is option 'C'. Can you explain this answer?.
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