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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 ...
**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?
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