Derivative of f(x) to find its maximum and minimum values

In summary, the conversation discusses finding the maximum and minimum values of a function with a positive constant, λ, and shows that the difference between them is 4(λ+λ^-1)^3. The conversation also mentions finding the least value of this difference as the parameter λ is varied. The solution involves expanding the right-hand side and using the quadratic formula to solve for x, taking into account the unknown value of λ.
  • #1
DryRun
Gold Member
838
4
Homework Statement
If λ is a positive constant, determine the maximum and minimum values of
f(x) = 9(4-3x^2)(λ-λ^-1-x)
and show that the difference between them is 4(λ+λ^-1)^3. Find the least value of this difference as the parameter λ is varied.

The attempt at a solution
I expanded the right-hand side and then did a first d.w.r.t.x

dy/dx = 81x^2 + 54x/λ - 54λx -36

I have to equate this to zero, to find either the minimum or maximum value:

81x^2 + 54x/λ - 54λx -36 = 0

But i don't know how to proceed next, since the value of λ is unknown, so i cannot solve for x.
 
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  • #2
You can certainly use the quadratic formula to solve for x. Of course those values, and the minimum and maximum values of the function, will depend upon [itex]\lamba[/itex].
 
  • #3
Mod note - moved from Precalc section.
 

FAQ: Derivative of f(x) to find its maximum and minimum values

What is the derivative of a function?

The derivative of a function is the rate of change of that function at a given point. It represents the slope of the tangent line to the function at that point.

How do you find the maximum and minimum values of a function using the derivative?

To find the maximum and minimum values of a function using the derivative, you first find the critical points of the function by setting the derivative equal to zero and solving for the values of x. Then, you evaluate the second derivative at each critical point. If the second derivative is positive, the critical point is a minimum value. If the second derivative is negative, the critical point is a maximum value.

Can a function have multiple maximum and minimum values?

Yes, a function can have multiple maximum and minimum values. This occurs when the graph of the function has multiple peaks and valleys. The number of maximum and minimum values depends on the complexity of the function and the number of critical points.

What is the difference between local and global maximum and minimum values?

A local maximum (or minimum) value is the highest (or lowest) point within a specific interval of the function, while a global maximum (or minimum) value is the highest (or lowest) point of the entire function. A global maximum (or minimum) value is also a local maximum (or minimum) value, but the converse is not always true.

Can a function have a maximum or minimum value at the endpoints of its domain?

Yes, a function can have a maximum or minimum value at the endpoints of its domain. This occurs when the domain of the function is a closed interval, and the endpoints are included in the function's range.

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