# How do you find the critical points to sketch the graph #h(x)=27x-x^3#?

##### 1 Answer

**Critical Points are:**

Graph of

#### Explanation:

**Given:**

We need to find the **Critical Points**

**Definition of Critical Points:**

Let **function** and let **a point in its domain**.

We call **Critical Point** if

(1)

(2) **undefined.**

Also not that,

(3) the derivative gives us the **slope of the tangent line**

(4) the **Critical Points** are points where the **slope of the tangent line is ZERO**

**Given:**

We will find the **derivative** first.

We are differentiating a polynomial. Hence, it is easy.

Note that

We will set this derivative equal to ZERO, to find our **Critical Points**.

Subtract

Divide both sides by

Divide both sides by 3

Taking **Square Root** on both sides

Hence our **Critical Points** are

Please **analyze the graph** below:

Please observe the following on the graph:

At the **Critical Points**, **tangent lines are horizontal** in this example