Find the Maximum/Minimum Value 18+2x-x^2
The given problem is an optimization question in calculus, specifically pertaining to finding the extreme values (maximum and/or minimum) of a quadratic function. The function in question is f(x) = 18 + 2x - x^2, which is expressed in standard polynomial form. This type of problem typically requires determining the vertex of the parabola represented by the quadratic equation, as the vertex corresponds to the extreme value. For quadratics in the form of f(x) = ax^2 + bx + c, the vertex can provide the maximum value of the function if a < 0 or the minimum value if a > 0.
$18 + 2 x - x^{2}$
Answer
Solution:
Step 1:
Identify the vertex of the parabola described by the quadratic equation $f(x) = ax^2 + bx + c$. The vertex's $x$-coordinate is given by $x_v = -\frac{b}{2a}$. For a parabola opening downwards (when $a < 0$), the vertex represents the maximum point.
Step 2:
Determine the $x$-coordinate of the vertex.
Step 2.1: Insert the known coefficients $a$ and $b$ into the vertex formula: $x_v = -\frac{2}{2(-1)}$.
Step 2.2: Simplify the expression by removing the parentheses: $x_v = -\frac{2}{-2}$.
Step 2.3: Further simplify the expression:
Step 2.3.1: Divide out the common factors: $x_v = \frac{1}{1}$.
Step 2.3.2: Simplify the fraction to find the vertex's $x$-coordinate: $x_v = 1$.
Step 3:
Calculate the maximum value of the function by evaluating $f(x)$ at $x_v$.
Step 3.1: Substitute $x_v$ into the original equation: $f(1) = 18 + 2(1) - (1)^2$.
Step 3.2: Simplify the equation to find the maximum value:
Step 3.2.1: Perform the multiplication: $f(1) = 18 + 2 - 1$.
Step 3.2.2: Combine the terms to get the final result: $f(1) = 19$.
Step 4:
The maximum value of the function is $19$, and it occurs at the vertex, which is at the point $(1, 19)$.
Knowledge Notes:
To find the maximum or minimum value of a quadratic function of the form $f(x) = ax^2 + bx + c$, one can use the vertex formula. The vertex of the parabola represented by the quadratic function is the point where the maximum or minimum value occurs, depending on the direction the parabola opens (upwards or downwards). The $x$-coordinate of the vertex is found using $x_v = -\frac{b}{2a}$, and the corresponding $y$-coordinate is $f(x_v)$. If $a > 0$, the parabola opens upwards and the vertex is the minimum point; if $a < 0$, the parabola opens downwards and the vertex is the maximum point.
In the given problem, the quadratic function is $f(x) = 18 + 2x - x^2$, which can be rewritten in standard form as $f(x) = -x^2 + 2x + 18$ (where $a = -1$, $b = 2$, and $c = 18$). Since $a$ is negative, the parabola opens downwards, and thus the vertex will give us the maximum value of the function.
The process of simplifying the vertex formula involves basic algebraic manipulation, including removing parentheses, canceling common factors, and simplifying fractions. Once the vertex's $x$-coordinate is found, it is substituted back into the original function to find the maximum (or minimum) value. Finally, the vertex point $(x_v, f(x_v))$ gives the location of the maximum or minimum on the graph of the function.
