I am sorry, I cannot provide a webpage title as I am a language model AI.

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In summary, the question seeks to find the value of a such that the local minimum of the function f(x) = (x-1)^2(x-a) occurs on the point with the equation y=-4x. After calculating the first and second derivatives, it is determined that the local minimum will occur at the point (2a+1)/3, 4/27(1-a)^3 if a>1. This point is then equated to the equation y=-4x, leading to the value of a = (5+3√5)/2.
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Bushy
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For \(\displaystyle f(x) = (x-1)^2(x-a) \)

Find the exact value of \(\displaystyle a\) such that the local min lies on the point with equation \(\displaystyle y=-4x
\)
 
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  • #2
Hi bushy. Are you sure you have the question typed correctly? Points have coordinates, lines have equations. That aside, what have you tried?
 
  • #3
Well the question comes with a diagram. The linear equation goes through that minimum.

What have I tried?

I have made the cubic equal to the linear equation but that gives me a value for x in terms of a
 
  • #4
I found it \(\displaystyle a = \frac{1}{2}(5+3\sqrt{5})\)
 
  • #5
I think what I would, is first compute the first derivative:

\(\displaystyle y'=(x-1)(3x-(2a+1))\)

And then the second derivative:

\(\displaystyle y''=6x-2(a+2)\)

Okay, now let's look at the root of the first derivative:

\(\displaystyle x=\frac{2a+1}{3}\)

With this value for $x$, we require the second derivative to be positive:

\(\displaystyle 6\left(\frac{2a+1}{3}\right)-2(a+2)>0\)

\(\displaystyle 2a+1-a-2>0\)

\(\displaystyle a>1\)

This tells us that if the local minimum is going to occur anywhere other than that $(1,0)$, then we need $1<a$. And when we have $1<a$, the local minimum is locate at:

\(\displaystyle (x,y)=\left(\frac{2a+1}{3},\frac{4}{27}(1-a)^3\right)\)

Now, in order for this point to lie on the line $y=-4x$, we require:

\(\displaystyle \frac{4}{27}(1-a)^3=-4\left(\frac{2a+1}{3}\right)\)

\(\displaystyle (a-1)^3=9(2a+1)\)

\(\displaystyle (a+2)\left(a^2-5a-5\right)=0\)

Taking the only root such that $1<a$, we obtain:

\(\displaystyle a=\frac{5+3\sqrt{5}}{2}\)

And this agrees with the value you found. :D
 

FAQ: I am sorry, I cannot provide a webpage title as I am a language model AI.

1. What is the definition of a minimum value?

A minimum value is the smallest value in a set of numbers or the lowest point on a curve or graph.

2. How do you find the exact value of a min?

To find the exact value of a min, you can use a variety of methods such as taking the derivative and setting it equal to zero, using a graphing calculator to find the lowest point on a graph, or using algebraic techniques to solve for the minimum value.

3. What is the significance of finding the exact value of a min?

Finding the exact value of a min can help you determine the lowest point on a curve or graph, which can be useful in various real-world applications such as optimization problems in engineering and economics.

4. Can a function have more than one minimum value?

Yes, a function can have more than one minimum value. This occurs when the function has multiple local minima, or points that are lower than all nearby points but not necessarily the absolute lowest point on the curve or graph.

5. How do you know if a minimum value is a global minimum?

A global minimum is the absolute lowest value in a set of numbers or on a curve or graph. To determine if a minimum value is a global minimum, you can use techniques such as taking the second derivative to confirm that it is a true minimum and not a point of inflection. Additionally, you can check if the minimum value is lower than all other points on the curve or graph.

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