Range of f(x): Find Max & Min Value with f'(x)

The minimum value of the range is negative infinity. In summary, the range of the function f(x)= ln(-x^2+x+6) is from negative infinity to the maximum value found by setting the derivative equal to zero. The minimum value of the range can be found by considering the function's domain, which must be between -2 and 3. As x approaches -2 from above or 3 from below, the function approaches 0 from above, indicating that the minimum value of the range is negative infinity.
  • #1
fiziksfun
78
0
Let f(x)= ln(-x^2+x+6)

Find the range of f(x). Use f'(x) to support your answer.

Attempt at a solution:

Find the max. value of the range is easy. I found the derivative and set it equal to zero. My REAL PROBLEM is finding the minimum value of the range.

The function's domain must be between -2 and 3 becus you cannot take the natural lg of a negative number or zero.
SO, as f(x) approaches, 3 or -2, does it approach negative infinity??
Am I right when I say its lower range is negative infinity?? HELP!
 
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  • #2
As x approaches -2 from above or 3 from below, then -x^2+x+6 approaches 0 from above. So yes, you are right.
 

What is the range of a function?

The range of a function is the set of all possible output values, or the y-values, that the function can produce. It can also be described as the set of numbers that the function maps to.

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

To find the maximum and minimum values of a function, you can use the derivative of the function. Set the derivative equal to zero and solve for the x-value. Then, plug this x-value into the original function to find the corresponding y-value, which will be the maximum or minimum value.

What does the derivative of a function represent?

The derivative of a function represents the rate of change of the function at a specific point. It tells us how much the output of the function is changing with respect to the input.

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

Yes, a function can have more than one maximum or minimum value. This can happen if the function has multiple peaks or valleys. The derivative of the function will be equal to zero at each of these points.

How do you determine if a maximum or minimum value is a global or local extremum?

A global extremum is the absolute maximum or minimum value of a function, meaning it is the largest or smallest value in the entire range of the function. A local extremum is a maximum or minimum value that is only the largest or smallest within a specific interval. To determine if a maximum or minimum value is a global or local extremum, you can look at the graph of the function or analyze the behavior of the function at different intervals.

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