Question

$f(x) = 2x^2 - 2x + 6$ will have a minimum value at $x=$ _____

• A. \( 0.2 \)
• B. \( 0.3 \)
• C. \( 0.4 \)
• D. \( 0.5 \)

Hint:

For a quadratic function $ax^2 + bx + c$ with $a>0$, the minimum occurs at $x = -b/(2a)$.

Answer 1

The Correct Option is D

The given function is a quadratic function of the form $f(x) = ax^2 + bx + c$, where $a = 2$, $b = -2$, and $c = 6$. Since $a>0$, the parabola opens upwards, and the function has a minimum value at its vertex. The x-coordinate of the vertex of a parabola $f(x) = ax^2 + bx + c$ is given by the formula $x = -\frac{b}{2a}$. Substituting the values of $a$ and $b$: $$x = -\frac{-2}{2(2)} = \frac{2}{4} = 0.5$$ Thus, the function $f(x) = 2x^2 - 2x + 6$ will have a minimum value at $x = 0.5$. Alternatively, we can use calculus. To find the minimum value, we take the first derivative of $f(x)$ and set it to zero: $$f'(x) = \frac{d}{dx}(2x^2 - 2x + 6) = 4x - 2$$ Set $f'(x) = 0$: $$4x - 2 = 0$$ $$4x = 2$$ $$x = \frac{2}{4} = 0.5$$ To confirm that this is a minimum, we take the second derivative: $$f''(x) = \frac{d}{dx}(4x - 2) = 4$$ Since $f''(0.5) = 4>0$, the function has a minimum at $x = 0.5$.