What is the Absolute Minimum Value of f(x)?

In summary, the absolute minimum value of the function f(x) = ln(x)/x is 1/e, obtained when the derivative is set to 0 and x is equal to e. This is based on the fact that taking the derivative and setting it to 0 results in the equation (1-ln(x))/x^2=0, which simplifies to x=e, and plugging this value into f(x) yields the minimum value of 1/e.
  • #1
karush
Gold Member
MHB
3,269
5
Let f be the function defined by
$f(x)=\dfrac{\ln x}{x}$ What is the absolute minimum value of f
a, 1
b. $\dfrac{1}{e}$
c. 0
d. e
e. none

ok I assume we take the derivative and then set it to zero

$\frac{1-\ln\left(x\right)}{x^{2}}=0$
$x=e$
 
Last edited:
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  • #2
karush said:
ok I assume we take the derivative and then set it to zero

then set that into f(x)

I think it is (B)

how do you know it’s not a minimum?
 
  • #3
karush said:
ok I assume we take the derivative and then set it to zero

then set that into f(x)

I think it is (B)
Okay, what do you get when you take the derivative? What x value makes that equal to 0? In other words, tell us why you "think it is (B)!
 

1. What is the definition of absolute max value?

The absolute max value is the largest possible value of a function or data set. It represents the highest point or peak on a graph.

2. How is the absolute max value different from the relative max value?

The absolute max value is the overall highest point on a graph, while the relative max value is the highest point within a specific interval. The relative max value can occur multiple times, while the absolute max value only occurs once.

3. How do you find the absolute max value of a function?

To find the absolute max value of a function, you must first take the derivative of the function and set it equal to 0. Then, solve for the x-value that makes the derivative equal to 0. Plug this x-value into the original function to find the corresponding y-value, which is the absolute max value.

4. Can a function have more than one absolute max value?

No, a function can only have one absolute max value. This is because the absolute max value represents the overall highest point on the graph, and there can only be one highest point.

5. What is the significance of the absolute max value in real-world applications?

The absolute max value is significant in real-world applications because it represents the maximum value of a function or data set. This can be useful in determining the maximum profit, maximum speed, or maximum capacity in a given situation.

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