# How do you find the maximum, minimum and inflection points for each function #f(x)= -14x^3 + 19x^3 - x - 2#?

Critical points occur at

First off, note that this is a third order function. A function of third order has no global maxima or minima; at one end it trends towards positive infinity, and at the other it trends towards negative infinity,. Therefore, no absolute minima or maxima exist. If this were a function of order 2

Local maxima and minima are types of critical points. Critical points are points where the function is neither increasing nor decreasing, i.e. where the slope of the line tangent to the function at that point is 0. Since the derivative of a function is the slope of its tangent line, that means that the critical points occur at the zeroes of the derivative.

Thus here:

Knowing that the non epsilon terms will cancel out..

Checking the second zero is left to the student, but you should be able to determine that it is a local maximum, meaning that the derivative changes from positive to negative as it crosses the 0.

Inflection points are the points where the second derivative is equal to 0. You can think of them as the critical points of the derivative , but this may be somewhat confusing.

Using the power rule;

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To find the maximum and minimum points, we first find the critical points by setting the derivative equal to zero and solving for (x). Then, we use the second derivative test to determine if each critical point is a maximum or minimum. For inflection points, we find where the second derivative equals zero and test the concavity of the function around those points. Let's start by finding the derivatives of the given function:

(f(x) = -14x^3 + 19x^2 - x - 2)

(f'(x) = -42x^2 + 38x - 1)

(f''(x) = -84x + 38)

Now, set the first derivative equal to zero and solve for (x) to find critical points:

(-42x^2 + 38x - 1 = 0)

Next, use the quadratic formula to solve for (x). Then, plug the values of (x) into the second derivative to classify each critical point as a maximum or minimum. Finally, for inflection points, set the second derivative equal to zero and solve for (x).

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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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