Find the Maximum/Minimum Value z=7x+9y

$z = 7 x + 9 y$

Step 1

Find the first derivative of the function.

By the Sum Rule, the derivative of $7 x + 9 y$ with respect to $x$ is $\frac{d}{d x} [7 x] + \frac{d}{d x} [9 y]$ .

$\frac{d}{d x} [7 x] + \frac{d}{d x} [9 y]$

Evaluate $\frac{d}{d x} [7 x]$ .

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$7 \frac{d}{d x} [x] + \frac{d}{d x} [9 y]$

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

$7 \cdot 1 + \frac{d}{d x} [9 y]$

Multiply $7$ by $1$ .

$7 + \frac{d}{d x} [9 y]$

Differentiate using the Constant Rule.

Since $9 y$ is constant with respect to $x$ , the derivative of $9 y$ with respect to $x$ is $0$ .

$7 + 0$

Add $7$ and $0$ .

$7$

Step 2

Since $7$ is constant with respect to $x$ , the derivative of $7$ 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.

$7 = 0$

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