The function ƒ(x) = 3x (x - 2) has:
minimum at x = 1
Analyzing the Quadratic Function
The given function is $f(x) = 3x(x - 2)$. This is a quadratic function.
First, let's expand the function to the standard quadratic form $ax^2 + bx + c$:
$$f(x) = 3x \times x - 3x \times 2$$ $$f(x) = 3x^2 - 6x$$In this form, we have $a = 3$, $b = -6$, and $c = 0$. Since the coefficient of the $x^2$ term ($a=3$) is positive, the parabola representing this function opens upwards. This means the function has a minimum value at its vertex.
Finding the Vertex of the Parabola
The x-coordinate of the vertex of a quadratic function in the form $f(x) = ax^2 + bx + c$ can be found using the formula:
$$x = -\frac{b}{2a}$$Substituting the values $a = 3$ and $b = -6$ into the formula:
$$x = -\frac{-6}{2 \times 3}$$ $$x = -\frac{-6}{6}$$ $$x = -(-1)$$ $$x = 1$$Therefore, the minimum value of the function $f(x) = 3x(x - 2)$ occurs at $x = 1$. This matches the first option.
Verifying the Minimum
We can also use calculus to find the minimum. First, find the derivative of the function:
$$f'(x) = \frac{d}{dx}(3x^2 - 6x)$$ $$f'(x) = 6x - 6$$To find critical points, set the derivative equal to zero:
$$6x - 6 = 0$$ $$6x = 6$$ $$x = 1$$Now, find the second derivative to determine if it's a minimum or maximum:
$$f''(x) = \frac{d}{dx}(6x - 6)$$ $$f''(x) = 6$$Since the second derivative $f''(1) = 6$ is positive, the function has a local minimum at $x = 1$. Because it's a parabola opening upwards, this local minimum is also the absolute minimum.
Conclusion
The function $f(x) = 3x(x - 2)$ has a minimum value at $x = 1$. This corresponds to the first option provided.
- The function is quadratic: $f(x) = 3x^2 - 6x$.
- The coefficient 'a' is positive ($a=3$), indicating a minimum.
- The minimum occurs at $x = -b / (2a) = -(-6) / (2 \times 3) = 1$.
Important Questions from Maxima & Minima
The maximum value attained by the function \(f\left( x \right) = x\left( {x - 1} \right)\left( {x - 2} \right)\) in the interval [1, 2] is _____.
- The minimum value of the function $f(x) = |x| + |2x + 3|$ for real $x$ is ________ (rounded off to 1 decimal place).
- The value of the determinant of the Hessian of $f(x,y) = x^2 + y^2 + xy - 8x - 7y$ at its stationary point is
- A rectangular box with an open top has an outer surface area of 12 m$^2$. The length of the base is twice the width. If the volume of the box is to be maximised, then the ratio of length to height should be ____________. (answer in integer)
- A wire of length $L$ is cut into two pieces to construct a circle and an equilateral triangle such that the combined area is minimum. The length of the wire used to construct the circle is
