# What are the local maxima and minima of #f(x) = 4x^3 + 3x^2 - 6x + 1#?

Polynomials are differentiable everywhere , so look for the critical values by simply finding the solutions to

Using algebra to solve this simple quadratic equation:

Determine if these are min or max by plugging into the second derivative:

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To find the local maxima and minima of ( f(x) = 4x^3 + 3x^2 - 6x + 1 ), first, find its derivative ( f'(x) ). Then, set ( f'(x) = 0 ) and solve for ( x ). The values of ( x ) obtained will correspond to critical points. Test these critical points using the second derivative test to determine whether they correspond to local maxima, minima, or points of inflection.

( f'(x) = 12x^2 + 6x - 6 )

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

( 12x^2 + 6x - 6 = 0 )

Solving this quadratic equation yields two solutions: ( x = -1 ) and ( x = \frac{1}{2} ).

To classify these critical points, find ( f''(x) ), the second derivative:

( f''(x) = 24x + 6 )

Evaluate ( f''(-1) ) and ( f''\left(\frac{1}{2}\right) ):

( f''(-1) = 24(-1) + 6 = -18 ) (negative, indicating concave down)

( f''\left(\frac{1}{2}\right) = 24\left(\frac{1}{2}\right) + 6 = 18 ) (positive, indicating concave up)

Since ( f''(-1) ) is negative, ( x = -1 ) corresponds to a local maximum, and since ( f''\left(\frac{1}{2}\right) ) is positive, ( x = \frac{1}{2} ) corresponds to a local minimum.

So, the local maximum is at ( x = -1 ) and the local minimum is at ( x = \frac{1}{2} ).

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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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