Find the Maximum/Minimum Value f(x)=2/3x^2+4x+1

$f (x) = \frac{2}{3} x^{2} + 4 x + 1$

Step 1

Combine $\frac{2}{3}$ and $x^{2}$ .

$f (x) = \frac{2 x^{2}}{3} + 4 x + 1$

Step 2

The minimum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is positive, the minimum value of the function is $f (- \frac{b}{2 a})$ .

$f_{min}$ $x = a x^{2} + b x + c$ occurs at $x = - \frac{b}{2 a}$

Step 3

Find the value of $x = - \frac{b}{2 a}$ .

Substitute in the values of $a$ and $b$ .

$x = - \frac{4}{2 (0.\overline{6})}$

Remove parentheses.

$x = - \frac{4}{2 (0.\overline{6})}$

Simplify $- \frac{4}{2 (0.\overline{6})}$ .

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$x = - \frac{2 \cdot 2}{2 \cdot 0.\overline{6}}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 0.\overline{6}$ .

$x = - \frac{2 \cdot 2}{2 (0.\overline{6})}$

Cancel the common factor.

$x = - \frac{2 \cdot 2}{2 \cdot 0.\overline{6}}$

Rewrite the expression.

$x = - \frac{2}{0.\overline{6}}$

Simplify the expression.

Divide $2$ by $0.\overline{6}$ .

$x = - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$x = - 3$

Step 4

Evaluate $f (- 3)$ .

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = \frac{2 {(- 3)}^{2}}{3} + 4 (- 3) + 1$

Simplify the result.

Simplify each term.

Raise $- 3$ to the power of $2$ .

$f (- 3) = \frac{2 \cdot 9}{3} + 4 (- 3) + 1$

Multiply $2$ by $9$ .

$f (- 3) = \frac{18}{3} + 4 (- 3) + 1$

Divide $18$ by $3$ .

$f (- 3) = 6 + 4 (- 3) + 1$

Multiply $4$ by $- 3$ .

$f (- 3) = 6 - 12 + 1$

Simplify by adding and subtracting.

Subtract $12$ from $6$ .

$f (- 3) = - 6 + 1$

Add $- 6$ and $1$ .

$f (- 3) = - 5$

The final answer is $- 5$ .

$- 5$

Step 5

Use the $x$ and $y$ values to find where the minimum occurs.

$(- 3, - 5)$

Step 6

Find the Maximum/Minimum Value f(x)=2/3x^2+4x+1

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