Let f(x) be a non-constant polynomial with real coefficients such that `f((1)/(2))=100` and f(x) `le` 100 for all real x. Which of the following statements is NOT necessarily true ?

Options

A. The coefficient of the highest degree term in f(x) is negative
B. f(x) has at least two real roots
C. If ` x ne 1//2` then ` f(x) lt 100`
D. At least one of the coefficients of f(x) is bigger than 50

Answer

To solve the problem, we need to analyze the given conditions about the polynomial \( f(x) \).

### Step 1: Understand the conditions
We know that:
1. \( f\left(\frac{1}{2}\right) = 100 \)
2. \( f(x) \leq 100 \) for all real \( x \)

### Step 2: Determine the nature of the polynomial
Since \( f(x) \) is a non-constant polynomial and it is bounded above by 100, the leading coefficient of \( f(x) \) must be negative. This is because if the leading coefficient were positive, \( f(x) \) would tend to \( +\infty \) as \( x \to +\infty \), contradicting the condition that \( f(x) \leq 100 \).

**Hint:** A polynomial that is bounded above must have a leading term with a negative coefficient.

### Step 3: Analyze the implications of the conditions
Given that \( f\left(\frac{1}{2}\right) = 100 \) is the maximum value of the polynomial, and \( f(x) \) approaches \( -\infty \) as \( x \) approaches \( +\infty \) or \( -\infty \), the polynomial must have at least two roots. This is because it must cross the x-axis at least twice to go from \( +\infty \) to \( -\infty \).

**Hint:** A polynomial that reaches a maximum value must have at least two roots if it goes from positive to negative.

### Step 4: Evaluate the options
Now, we need to evaluate the provided options to find which one is NOT necessarily true.

1. **Option 1:** The leading coefficient is negative. (True)
2. **Option 2:** \( f(x) \) has at least two roots. (True)
3. **Option 3:** If \( f(x) \neq 100 \), then \( f(x) < 100 \). (Not necessarily true, as shown by the counterexample.)
4. **Option 4:** At least one coefficient of \( f(x) \) is greater than 50. (True)

### Step 5: Identify the false statement
From the analysis, we see that **Option 3** is not necessarily true because there can be values of \( x \) (like \( x = -\frac{1}{2} \)) where \( f(x) \) can equal 100 without violating the conditions.

**Final Answer:** The statement that is NOT necessarily true is **Option 3**.