Find the Maximum/Minimum Value z=3x+2y

$z = 3 x + 2 y$

Step 1

Find the first derivative of the function.

By the Sum Rule, the derivative of $3 x + 2 y$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [2 y]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [2 y]$

Evaluate $\frac{d}{d x} [3 x]$ .

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x] + \frac{d}{d x} [2 y]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 \cdot 1 + \frac{d}{d x} [2 y]$

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [2 y]$

Differentiate using the Constant Rule.

Since $2 y$ is constant with respect to $x$ , the derivative of $2 y$ with respect to $x$ is $0$ .

$3 + 0$

Add $3$ and $0$ .

$3$

Step 2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f'' (x) = 0$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$3 = 0$

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema

Step 6