How do you find the critical points to sketch the graph #h(x)=27xx^3#?
Critical Points are:
Graph of
Given:
We need to find the Critical Points
Definition of Critical Points:
Let
We call
(1)
(2)
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
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To find the critical points of the function ( h(x) = 27x  x^3 ) in order to sketch its graph, follow these steps:

Compute the derivative of ( h(x) ) to find critical points: [ h'(x) = \frac{d}{dx}(27x  x^3) ] [ h'(x) = 27  3x^2 ]

Set the derivative equal to zero and solve for ( x ) to find critical points: [ 27  3x^2 = 0 ] [ 3x^2 = 27 ] [ x^2 = 9 ] [ x = \pm 3 ]

Determine the nature of critical points by analyzing the sign of the derivative around each critical point.

For ( x = 3 ):
 Substitute ( x = 3 ) into ( h'(x) ): ( h'(3) = 27  3(3)^2 = 27  27 = 0 ).
 Since the derivative changes sign from negative to positive at ( x = 3 ), it indicates a local minimum.

For ( x = 3 ):
 Substitute ( x = 3 ) into ( h'(x) ): ( h'(3) = 27  3(3)^2 = 27  27 = 0 ).
 Since the derivative changes sign from positive to negative at ( x = 3 ), it indicates a local maximum.


Plot the critical points ( (3, h(3)) ) and ( (3, h(3)) ) on the graph.

Determine the behavior of the function as ( x ) approaches positive and negative infinity:
 As ( x ) approaches positive infinity, ( h(x) ) approaches negative infinity because of the term ( x^3 ).
 As ( x ) approaches negative infinity, ( h(x) ) approaches positive infinity because the term ( 27x ) dominates.

Sketch the graph based on the behavior around critical points and at infinity.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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