Continuity Of A Function

Step 1: Analyze the function on the interval .
The function is a parabola opening upwards.
At , .
At , .
Thus, the maximum value occurs at both and .

Step 2: Check each option.
\begin{itemize} \item (A) has a minimum at . False, minimum is at but the statement about maximum needed. \item (B) has the maximum at . True, . \item (C) is continuous on . True, is continuous everywhere. \item (D) is bounded on . True, bounded between 0 and 1. \end{itemize}

Step 1: Expand .
Given , so
Step 2: Expand .
Also, thus
Step 3: Compare coefficients from both sides.
From equating the coefficients: Coefficient of : (ok).
Coefficient of : (ok).
Coefficient of : .
Coefficient of : (ok).
Coefficient of : (ok).
Constant term: .

Step 4: Find using the given condition.
Given: Substituting and ,
Step 5: Find .
Now,

We are given , and we need to find .
Step 1: Compute .
To find , we first substitute into itself. Thus, we compute: Simplifying both the numerator and denominator: So, we have:
Step 2: Compute .
Now that we know , we can apply again: Thus, the value of is , which is the same as .
Step 3: Conclusion.
The answer is , which matches option (D).