Expression `4x^(2) - 16x + 17` will be minimum when x= ?

Answer

To find the value of \( x \) at which the expression \( 4x^2 - 16x + 17 \) reaches its minimum, we can follow these steps:

### Step 1: Identify the Expression
The given expression is:
\[ f(x) = 4x^2 - 16x + 17 \]

### Step 2: Differentiate the Expression
To find the minimum value, we need to differentiate the expression with respect to \( x \):
\[ f'(x) = \frac{d}{dx}(4x^2) - \frac{d}{dx}(16x) + \frac{d}{dx}(17) \]

### Step 3: Apply the Power Rule
Using the power rule of differentiation, we compute:
- The derivative of \( 4x^2 \) is \( 8x \).
- The derivative of \( -16x \) is \( -16 \).
- The derivative of a constant \( 17 \) is \( 0 \).

Thus, we have:
\[ f'(x) = 8x - 16 \]

### Step 4: Set the Derivative to Zero
To find the critical points, we set the derivative equal to zero:
\[ 8x - 16 = 0 \]

### Step 5: Solve for \( x \)
Now, we solve for \( x \):
\[ 8x = 16 \]
\[ x = \frac{16}{8} = 2 \]

### Step 6: Confirm Minimum Value
To confirm that this point is indeed a minimum, we can check the second derivative:
\[ f''(x) = \frac{d}{dx}(8x - 16) = 8 \]
Since \( f''(x) = 8 > 0 \), this indicates that the function is concave up at \( x = 2 \), confirming that it is a minimum point.

### Final Answer
The expression \( 4x^2 - 16x + 17 \) will be minimum when:
\[ x = 2 \]

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