Find the Maximum/Minimum Value 2-3x^2
The given problem is asking to determine the highest or lowest possible value of the mathematical expression 2 - 3x^2, where x is a variable. This is an optimization problem where one needs to identify the maximum or the minimum point on the graph of the given quadratic equation.
$2 - 3 x^{2}$
Answer
Solution:
Step:1 To determine the extremum of a parabola described by $f(x) = ax^2 + bx + c$, we use the vertex formula $x = -\frac{b}{2a}$. For $a < 0$, this gives the maximum value at $f(-\frac{b}{2a})$.
Step:2 Calculate the vertex $x$-coordinate using $x = -\frac{b}{2a}$.
Step:2.1 Insert the given coefficients for $a$ and $b$: $x = -\frac{0}{2(-3)}$.
Step:2.2 Eliminate the brackets: $x = -\frac{0}{2(-3)}$.
Step:2.3 Simplify the expression $-\frac{0}{2(-3)}$.
Step:2.3.1 Identify and remove any common factors between numerator and denominator.
Step:2.3.1.1 Extract the factor of $2$ from the numerator: $x = -\frac{2(0)}{2(-3)}$.
Step:2.3.1.2 Eliminate the common factors.
Step:2.3.1.2.1 Remove the common factor of $2$: $x = -\frac{\cancel{2} \cdot 0}{\cancel{2} \cdot -3}$.
Step:2.3.1.2.2 Rewrite the simplified expression: $x = -\frac{0}{-3}$.
Step:2.3.2 Remove any common factors between $0$ and $-3$.
Step:2.3.2.1 Factor out $3$ from the numerator: $x = -\frac{3(0)}{-3}$.
Step:2.3.2.2 Transfer the negative sign from the denominator: $x = -(-1 \cdot 0)$.
Step:2.3.3 Express $-1 \cdot 0$ as $0$: $x = -0$.
Step:2.3.4 Perform the multiplication of $-0$.
Step:2.3.4.1 Multiply $-1$ by $0$: $x = 0$.
Step:2.3.4.2 Confirm the result of the multiplication: $x = 0$.
Step:3 Compute $f(0)$.
Step:3.1 Substitute $0$ for $x$ in the function: $f(0) = 2 - 3(0)^2$.
Step:3.2 Simplify the expression.
Step:3.2.1 Break down and simplify each term.
Step:3.2.1.1 Zero raised to any power is $0$: $f(0) = 2 - 3 \cdot 0$.
Step:3.2.1.2 Multiply $-3$ by $0$: $f(0) = 2 + 0$.
Step:3.2.2 Combine $2$ and $0$: $f(0) = 2$.
Step:3.2.3 The maximum value is $2$.
Step:4 Identify the coordinates of the maximum point: $(0, 2)$.
Step:5
The maximum value of the function $2 - 3x^2$ is $2$, which occurs at the point $(0, 2)$.
Knowledge Notes:
Quadratic Functions: A quadratic function is of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a \neq 0$. The graph of a quadratic function is a parabola.
Vertex of a Parabola: The vertex of a parabola $y = ax^2 + bx + c$ is the point where the function attains its maximum or minimum value. The $x$-coordinate of the vertex is given by $x = -\frac{b}{2a}$.
Maximum and Minimum Values: For a quadratic function $f(x) = ax^2 + bx + c$, if $a > 0$, the parabola opens upwards, and the vertex represents the minimum value. If $a < 0$, the parabola opens downwards, and the vertex represents the maximum value.
Simplifying Expressions: When simplifying expressions, especially those involving zero, remember that any number multiplied by zero equals zero, and zero divided by any nonzero number is also zero.
