How do you use the first and second derivatives to sketch #y=(x^3)(6x^2)+5x+12#?
The first derivative allows you to determine the location of the critical points of the function. The second allows you to determine the nature of these points.
Using the quadratic formula:
This gives the location of the turning points, we now find the second derivative to determine their natures.
We can see that for x = 3.528 that y'' > 0, x = 0.472 that y'' < 0. For positive curvature, it will be curving upwards (think a positive, happy smiley face :)) so the point there will be in a trough, ie a minimum. The point with negative curvature will be curving downwards so will be a maximum.
With this, plug in the x values to find the values of the function at these critical points, mark them taking note of their natures and then, taking into account other useful features like intercepts and asymptotes, plot the graph.
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To sketch the curve ( y = x^3  6x^2 + 5x + 12 ) using the first and second derivatives:

Find the first derivative ( y' ): [ y' = 3x^2  12x + 5 ]

Set ( y' = 0 ) to find critical points: [ 3x^2  12x + 5 = 0 ] Solve for ( x ) to find the xvalues of the critical points.

Use the second derivative test to determine the nature of the critical points: [ y'' = 6x  12 ]
 If ( y'' > 0 ), the function is concave up at that point (local minimum).
 If ( y'' < 0 ), the function is concave down at that point (local maximum).
 If ( y'' = 0 ), the test is inconclusive.
 Sketch the curve:
 Plot the critical points and the points where the first derivative is zero.
 Use the information from the second derivative test to determine the concavity between the critical points.
 Sketch the curve passing through these points, considering the concavity and the behavior of the function as ( x ) approaches infinity or negative 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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