Determine the location and nature of turning point

In summary, the turning point of (1-lnx)^2, x>2 is a minimum and the derivative can be found using the chain rule.
  • #1
markosheehan
136
0
whats the turning point of (1-lnx)², x>2 is it a minimum or maximum. can someone help me
 
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  • #2
Re: calculus

We have \(\displaystyle (1-\ln x)^2\ge0\) and, moreover, $(1-\ln x)^2=0\iff \ln x=1$. Can you find the solution of $\ln x =1$?
 
  • #3
how do you get the derivative of it
 
  • #4
As for any composition of functions. The outer function if $f(x)=x^2$, so first you differentiate $x^2$, and the result is $2x$. Now our whole function is not $f(x)$, but $f(1-\ln x)$. Therefore you replace $x$ in $2x$ with $1-\ln x$ to get $2(1-\ln x)$. To finish, you need to multiply this by the derivative of $1-\ln x$.

I recommend reviewing the chain rule for computing the derivative of the composition of functions in your textbook.
 

FAQ: Determine the location and nature of turning point

What is a turning point?

A turning point is a point on a graph where the direction of the curve changes, from increasing to decreasing or vice versa.

How do you determine the location of a turning point?

To determine the location of a turning point, you need to find the x-value where the slope of the curve is equal to zero. This can be done by finding the derivative of the function and setting it equal to zero, then solving for x.

What does the nature of a turning point refer to?

The nature of a turning point refers to whether the point is a local maximum or minimum. A local maximum is the highest point on a curve within a specific interval, while a local minimum is the lowest point on a curve within a specific interval.

How do you determine the nature of a turning point?

To determine the nature of a turning point, you need to look at the second derivative of the function at that point. If the second derivative is positive, the turning point is a local minimum. If the second derivative is negative, the turning point is a local maximum. If the second derivative is zero, the nature of the turning point is indeterminate.

Why is it important to determine the location and nature of turning points?

Determining the location and nature of turning points can help us understand the behavior of a function and identify important features such as maxima and minima. This information is useful in various fields such as economics, physics, and engineering, where the analysis of curves and functions is crucial in making accurate predictions and decisions.

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