Find the minimum value of the function $$ f(x)=2 x^{2}-6.7 x+2 $$ to the nearest hundredth.

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UpStudy AI · Accepted Answer

To find the minimum value of the quadratic function $$ f(x) = 2x^2 - 6.7x + 2 $$, we can follow these steps:

1. **Determine the Vertex:**
   - For a quadratic function of the form $$ f(x) = ax^2 + bx + c $$, the vertex (which gives the minimum or maximum value) occurs at $$ x = -\frac{b}{2a} $$.
   - Here, $$ a = 2 $$ and $$ b = -6.7 $$.
   $$
   x = -\frac{-6.7}{2 \times 2} = \frac{6.7}{4} = 1.675
   $$

2. **Calculate the Function Value at the Vertex:**
   - Substitute $$ x = 1.675 $$ into the function:
   $$
   f(1.675) = 2(1.675)^2 - 6.7(1.675) + 2
   $$
   - First, calculate $$ (1.675)^2 $$:
     $$
     1.675^2 = 2.805625
     $$
   - Multiply by 2:
     $$
     2 \times 2.805625 = 5.61125
     $$
   - Calculate $$ -6.7 \times 1.675 $$:
     $$
     -6.7 \times 1.675 = -11.2225
     $$
   - Add the constants:
     $$
     5.61125 - 11.2225 + 2 = -3.61125
     $$

3. **Round to the Nearest Hundredth:**
   $$
   -3.61125 \approx -3.61
   $$

**Minimum Value:** $$-3.61$$
The minimum value of the function $$ f(x) = 2x^2 - 6.7x + 2 $$ is $$-3.61$$.

Find the minimum value of the function to the nearest hundredth.

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