Question

If a quadratic function in \( x \) has the value 19 when \( x = 1 \) and has a maximum value 20 when \( x = 2 \), then the function is

• A. \( f(x) = x^2 - 4x + 16 \)
• B. \( f(x) = -x^2 + 5x + 16 \)
• C. \( f(x) = x^2 + 4x + 16 \)
• D. \( f(x) = -x^2 + 4x + 16 \)

Hint:

For a quadratic function, negative coefficient of \(x^2\) means maximum value and positive coefficient means minimum value.

Answer 1

The Correct Option is D

Step 1: Use maximum value condition.
A quadratic function has maximum value when its coefficient of \(x^2\) is negative.
So options with negative coefficient of \(x^2\) are possible.

Step 2: Check option (B).
\[ f(x) = -x^2 + 5x + 16 \]
At \(x = 2\):
\[ f(2) = -4 + 10 + 16 = 22 \]
So option (B) is not correct.

Step 3: Check option (D).
\[ f(x) = -x^2 + 4x + 16 \]

Step 4: Verify value at \(x = 1\).
\[ f(1) = -(1)^2 + 4(1) + 16 \]
\[ = -1 + 4 + 16 = 19 \]

Step 5: Verify maximum value at \(x = 2\).
\[ f(2) = -(2)^2 + 4(2) + 16 \]
\[ = -4 + 8 + 16 = 20 \]

Step 6: Confirm maximum nature.
Since coefficient of \(x^2\) is negative, the parabola opens downward.
Therefore, \(20\) is the maximum value.

Step 7: Final conclusion.
\[ \boxed{f(x) = -x^2 + 4x + 16} \]