How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $F(x)= (x^2)(lnx)$ ?

Question

How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $F(x)= (x^2)(lnx)$ ?

1 Answer

Impact of this question

1802 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-01-03T10:23:21

By analysing $F'(x)$ over the domain of $F(x): x>0$
$F(x)$ has an absolute minimum at $x=1/sqrt(e)$
$F(x)$ is decreasing for $0< x<1/sqrt(e)$ and increasing for $x>1/sqrt(e)$

Explanation:

$F(x) = x^2lnx$

$F'(x) = x^2.!/x + lnx*2x$ (Product Rule)

$= x+2x*lnx$

For local extrema $F'(x) =0$

Thus: $x+2xlnx = 0$

$x(1+2lnx)=0$

$x=0$ or $lnx=-1/2$

Since $F(x)$ is defined for $x>0$ our local extrema will be at:

$lnx=-1/2 -> x=e^(-1/2) = 1/sqrt(e) ~= 0.60653$

Hence:
$F(x)$ has an absolute minimum at $x=1/sqrt(e)$
$F(x)$ is decreasing for $x<1/sqrt(e)$ and increasing for $x>1/sqrt(e)$

This is shown by the graph of $F(x)$ below:

graph{x^2*lnx [-0.542, 2.877, -0.449, 1.26]}

Science

Math

Humanities

... and beyond

How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $F(x)= (x^2)(lnx)$ ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for F(x)= (x^2)(lnx)F(x)= (x^2)(lnx)?

1 Answer

Explanation:

Related questions

Impact of this question

How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $F(x)= (x^2)(lnx)$ ?