Maximum - Can this be solved algebraically?

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Second Derivative Test to verify the max if the Newtons Method was successfulIn summary, the conversation discusses finding the absolute extrema of the function f(x) = e^-x * ln(lnx). The person attempted to find the extrema by taking the first derivative and setting it to zero, but encountered difficulty in solving for x without a graphing calculator. They also mention alternative forms of the equation and suggest the use of numerical methods. Another person chimes in and suggests using Newton's Method and the Second Derivative Test to verify the maximum.
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turdferguson
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Homework Statement


Find the absolute extrema of f(x) = e^-x * ln(lnx)

The Attempt at a Solution


Ive successfully taken the first derivative and set it to zero. The problem is checking the sign of 1/(xlnx) - ln(lnx)

No matter how I try to manipulate this, I can't seem to isolate x. Its clear with a graphing calculator that f' changes from positive to negative at about 3.5, but this is supposed to be done without a graphing utility (if I could use one in my answer, I might as well have found the maximum from the start by graphing). Can this be set to zero and solved without a calculator?

Other forms include
1 - x*lnx*ln(lnx) = 0
lnx^(xlnx) - e = 0
 
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  • #2
No, I don't think you can solve that without using numerical methods, such as a graphing calculator.
 
  • #3
I found out today that we were supposed to use Newtons Method with a scientific calculator
 

1. Can the maximum of a function be found algebraically?

Yes, the maximum of a function can be found algebraically by taking the derivative of the function and setting it equal to zero. Then, solving for the variable will give the x-coordinate of the maximum point. Substituting this value back into the original function will give the y-coordinate of the maximum point.

2. Is there a formula for finding the maximum of a set of numbers?

Yes, the formula for finding the maximum of a set of numbers is simply the largest number in the set. This can be found by ordering the numbers from least to greatest and selecting the last number in the list.

3. Are there any limitations to solving for the maximum algebraically?

Yes, there are limitations to solving for the maximum algebraically. Depending on the complexity of the function, it may be difficult or impossible to find the maximum using algebraic methods. In these cases, numerical methods such as graphing or using a calculator may be necessary.

4. Can the maximum of a function change?

Yes, the maximum of a function can change depending on the values of the input variables. For example, if the function is a polynomial, changing the coefficients or the degree of the polynomial can change the location of the maximum point.

5. What is the difference between the maximum and the global maximum of a function?

The maximum of a function refers to the highest point on a specific interval, while the global maximum refers to the highest point of the entire function. The global maximum may be found at a different point than the local maximum, depending on the shape of the function.

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