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

To find the minimum value of the quadratic function \( f(x)=2 x^{2}-6.7 x+2 \), we can use the vertex formula for a parabola given by \( f(x) = ax^2 + bx + c \), where the x-coordinate of the vertex (which gives us the minimum for \( a > 0 \)) is found using \( x = -\frac{b}{2a} \). Here, \( a = 2 \) and \( b = -6.7 \). Thus, \[ x = -\frac{-6.7}{2 \cdot 2} = \frac{6.7}{4} = 1.675 \] Next, we will find the corresponding minimum value of \( f \) by substituting \( x = 1.675 \) back into the function: \[ f(1.675) = 2(1.675)^2 - 6.7(1.675) + 2 \] Calculating \( (1.675)^2 \): \[ (1.675)^2 \approx 2.803625 \] Now we can substitute this back into the function: \[ f(1.675) = 2(2.803625) - 6.7(1.675) + 2 \] \[ = 5.60725 - 11.2225 + 2 \] \[ = 5.60725 - 11.2225 + 2 \approx 5.60725 - 11.2225 + 2 \approx -3.61525 \] Now rounding to the nearest hundredth: \[ -3.61525 \approx -3.62 \] Therefore, the minimum value of the function \( f(x) \) to the nearest hundredth is \[ \boxed{-3.62} \]