Finding the x-value of Max, Min, & Inflexion Point

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In summary, for part i) the first and second derivatives do not share a common root, and for part iii) we use the equation f'(x)=4ax^2-2bx=2x(2ax-b)=0 to find the critical values. The $x$ value of the local minimum is x=\frac{b}{2a} and the $x$-value of the point of inflection is x=\frac{b}{4a}. This information can be used to answer part iv).
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markosheehan
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i know how to do the first two parts the x value of the max is 0 and of the minimum is 8 and 4 is the x value of the inflexion point. however i don't know how to do part 3
 

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Well, for part i) this can be explained using the fact that the first and second derivatives to not share a common root. I agree with your assessment of the $x$-values for the local extrema and point of inflection.

To answer part iii), we need to begin with:

\(\displaystyle f(x)=ax^3-bx^2\)

To find the critical values, we need to compute $f;(x)$ and equate this to zero and solve for $x$:

\(\displaystyle f'(x)=4ax^2-2bx=2x(2ax-b)=0\)

Hence:

\(\displaystyle x\in\left\{0,\frac{b}{2a}\right\}\)

From the graph, we know that the $x$ value of the local minimum must be:

\(\displaystyle x=\frac{b}{2a}\)

For the point of inflection, we set $f''(x)=0$:

\(\displaystyle f''(x)=8ax-2b=2(4a-b)=0\)

Hence, the $x$-value of the point of inflection must be:

\(\displaystyle x=\frac{b}{4a}\)

And now we have enough information to answer part iv). :D
 

FAQ: Finding the x-value of Max, Min, & Inflexion Point

What is the process for finding the x-value of a maximum point?

To find the x-value of a maximum point, you first need to find the derivative of the function. Then, set the derivative equal to zero and solve for the x-value. This x-value will be the x-coordinate of the maximum point.

How do you determine the x-value of a minimum point?

Similar to finding the x-value of a maximum point, you need to find the derivative of the function and set it equal to zero. However, this time you need to look for the x-value that makes the second derivative positive. This x-value will be the x-coordinate of the minimum point.

What is the significance of finding the x-value of an inflexion point?

An inflexion point is where the concavity of a function changes. The x-value of this point can provide information about the overall behavior of the function. For example, if the inflexion point has a positive x-value, it means the function is concave up on that interval.

Can a function have more than one maximum or minimum point?

Yes, a function can have multiple maximum or minimum points. This can happen if the function has multiple local extrema or if it is a periodic function with repeating maximum and minimum values.

Is there a shortcut method for finding the x-value of a maximum, minimum, or inflexion point?

While there are some techniques such as the first and second derivative tests that can aid in finding these points, there is no shortcut method. The most accurate way to find the x-value is by using the derivative and solving for zero.

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