If `f(x)=1+2x^2+4x^4+6x^6+....+100x^100` is a polynomial in 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

Options

A. neither a maximum nor a minimum
B. only one maximum
C. only one minimum
D. one maximum and one minimum

Answer

To solve the problem, we need to analyze the function \( f(x) = 1 + 2x^2 + 4x^4 + 6x^6 + \ldots + 100x^{100} \).

### Step 1: Rewrite the function
The function can be expressed as:
\[ f(x) = \sum_{n=0}^{50} (2n)x^{2n} = 1 + 2x^2 + 4x^4 + 6x^6 + \ldots + 100x^{100} \]

### Step 2: Differentiate the function
We differentiate \( f(x) \) with respect to \( x \):
\[ f'(x) = 0 + 2 \cdot 2x + 4 \cdot 4x^3 + 6 \cdot 6x^5 + \ldots + 100 \cdot 100x^{99} \]
This simplifies to:
\[ f'(x) = 4x + 16x^3 + 36x^5 + \ldots + 10000x^{99} \]

### Step 3: Factor out \( x \)
We can factor out \( x \) from \( f'(x) \):
\[ f'(x) = x(4 + 16x^2 + 36x^4 + \ldots + 10000x^{98}) \]

### Step 4: Analyze the critical points
Setting \( f'(x) = 0 \):
\[ x(4 + 16x^2 + 36x^4 + \ldots + 10000x^{98}) = 0 \]
This gives us:
1. \( x = 0 \)
2. The term \( (4 + 16x^2 + 36x^4 + \ldots + 10000x^{98}) \) must be analyzed.

### Step 5: Analyze the second term
The expression \( 4 + 16x^2 + 36x^4 + \ldots + 10000x^{98} \) consists of squares and is always non-negative. Since the lowest term is \( 4 \), this expression is always greater than zero for all \( x \neq 0 \).

### Step 6: Determine the nature of the critical point
At \( x = 0 \):
- \( f'(x) \) changes sign:
- For \( x < 0 \), \( f'(x) < 0 \) (decreasing).
- For \( x > 0 \), \( f'(x) > 0 \) (increasing).

Thus, \( x = 0 \) is a minimum point.

### Conclusion
Since \( f(x) \) has only one critical point at \( x = 0 \) and it is a minimum, the correct option is:
(c) only one minimum.