Find the maximum or minimum value of the function. $$ f(x)=-\frac{x^{2}}{3}+2 x+6 $$ Is this a maximum or minimum value? $$ \bigcirc $$ maximum value minimum value

Question

UpStudy AI · Accepted Answer

1. The given function is
   $$
   f(x)=-\frac{x^{2}}{3}+2x+6.
   $$

2. Since the coefficient of $$ x^2 $$ is $$-\frac{1}{3}$$ (which is negative), the parabola opens downward. This means that the vertex of the parabola represents a maximum value.

3. The vertex of a quadratic function of the form
   $$
   f(x)=ax^2+bx+c
   $$
   is given by
   $$
   x=-\frac{b}{2a}.
   $$
   For the given function, $$ a=-\frac{1}{3} $$ and $$ b=2 $$.

4. Calculate the $$ x $$-coordinate of the vertex:
   $$
   x=-\frac{2}{2\left(-\frac{1}{3}\right)}=-\frac{2}{-\frac{2}{3}}=3.
   $$

5. Substitute $$ x=3 $$ back into the function to find the $$ y $$-coordinate (maximum value):
   $$
   f(3)=-\frac{(3)^2}{3}+2(3)+6.
   $$
   $$
   f(3)=-\frac{9}{3}+6+6.
   $$
   $$
   f(3)=-3+6+6=9.
   $$

6. Therefore, the function attains a maximum value of $$ 9 $$ at $$ x=3 $$.

$$
\textbf{Maximum value: } 9.
$$
The function has a maximum value of 9 at $$ x=3 $$.

Find the maximum or minimum value of the function. \[ f(x)=-\frac{x^{2}}{3}+2 x+6 \] Is this a maximum or minimum value? \( \bigcirc \) maximum value minimum value

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Calculus Questions