# How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #f(x)= 3x^4+4x^3-12x^2+5#?

Tthe function has a maximum at

The function has a minimum at

The function has a minimum at

Given -

find the first two derivatives

Set first derivative equal to zero

graph{3x^4+4x^3-12x^2+5 [-10, 10, -5, 5]} #

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To determine where the function is increasing or decreasing, and where relative maxima and minima occur for ( f(x) = 3x^4 + 4x^3 - 12x^2 + 5 ):

- Find the first derivative of the function.
- Set the first derivative equal to zero to find critical points.
- Use the first derivative test to determine the intervals where the function is increasing or decreasing.
- Find the second derivative of the function.
- Use the second derivative test to determine where the relative maxima and minima occur.

First Derivative: ( f'(x) = 12x^3 + 12x^2 - 24x )

Critical Points: Set ( f'(x) = 0 ) and solve for ( x ).

Second Derivative: ( f''(x) = 36x^2 + 24x - 24 )

Use the Second Derivative Test:

- If ( f''(x) > 0 ), the function is concave up, indicating a relative minimum.
- If ( f''(x) < 0 ), the function is concave down, indicating a relative maximum.

Determine intervals of increase and decrease using the First Derivative Test:

- If ( f'(x) > 0 ), the function is increasing.
- If ( f'(x) < 0 ), the function is decreasing.

Combine the information from steps 3 and 5 to locate relative maxima and minima.

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When evaluating a one-sided 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 one-sided 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 one-sided 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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