Question

For the quadratic function \(f(x) = x^2 - kx + 12\), the minimum value of \(f(x)\) is 3. Which of the following is the value of \(k\)?

• A. 6
• B. 8
• C. 10
• D. 12

Hint:

A quick way to find the minimum/maximum value of a quadratic function \(ax^2 + bx + c\) is to use the formula \(\frac{4ac - b^2}{4a}\). For this problem, setting \(\frac{4(1)(12) - (-k)^2}{4(1)} = 3\) gives \(48 - k^2 = 12\), which quickly leads to \(k^2 = 36\).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given a quadratic function \(f(x) = ax^2 + bx + c\), where \(a=1\), \(b=-k\), and \(c=12\). Since the coefficient of \(x^2\) is positive (\(a=1>0\)), the parabola opens upwards, and the function has a minimum value. We are given that this minimum value is 3, and we need to find the value of the parameter \(k\).
Step 2: Key Formula or Approach:
The minimum value of a quadratic function \(f(x) = ax^2 + bx + c\) occurs at its vertex. The x-coordinate of the vertex is given by the formula:
\[ x = -\frac{b}{2a} \] The minimum value is the value of the function at this x-coordinate, i.e., \(f(-\frac{b}{2a})\).
Step 3: Detailed Explanation:
For the given function \(f(x) = x^2 - kx + 12\), we have:
- \(a = 1\)
- \(b = -k\)
- \(c = 12\)
First, we find the x-coordinate where the minimum value occurs:
\[ x = -\frac{-k}{2(1)} = \frac{k}{2} \] Next, we substitute this value of \(x\) back into the function to find the minimum value in terms of \(k\):
\[ f\left(\frac{k}{2}\right) = \left(\frac{k}{2}\right)^2 - k\left(\frac{k}{2}\right) + 12 \] \[ f\left(\frac{k}{2}\right) = \frac{k^2}{4} - \frac{k^2}{2} + 12 \] To combine the fractions, we find a common denominator:
\[ f\left(\frac{k}{2}\right) = \frac{k^2 - 2k^2}{4} + 12 \] \[ f\left(\frac{k}{2}\right) = -\frac{k^2}{4} + 12 \] We are given that the minimum value of \(f(x)\) is 3. Therefore, we can set our expression for the minimum value equal to 3:
\[ -\frac{k^2}{4} + 12 = 3 \] Now, we solve for \(k\):
\[ 12 - 3 = \frac{k^2}{4} \] \[ 9 = \frac{k^2}{4} \] \[ k^2 = 9 \times 4 \] \[ k^2 = 36 \] \[ k = \pm\sqrt{36} \] \[ k = \pm 6 \] The options provided are all positive values. Therefore, the correct value for \(k\) from the given choices is 6.
Step 4: Final Answer
The value of \(k\) is 6.