Find the Maximum/Minimum Value 2-3x^2

Solution:

Step:1 To determine the extremum of a parabola described by $f(x)=a{x}^2+bx+c$, 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:

1. Quadratic Functions: A quadratic function is of the form $f(x)=a{x}^2+bx+c$, where $a$, $b$, and $c$ are constants, and $a\ne 0$. The graph of a quadratic function is a parabola.

2. Vertex of a Parabola: The vertex of a parabola $y=a{x}^2+bx+c$ is the point where the function attains its maximum or minimum value. The $x$-coordinate of the vertex is given by $x=-\frac{b}{2a}$.

3. Maximum and Minimum Values: For a quadratic function $f(x)=a{x}^2+bx+c$, 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.

4. 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.