Home > Calculus > Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)

Is #f(x)=x^2lnx# increasing or decreasing at #x=1#?

Answer 1

Evelyn Agard

Increasing.

You can tell if a function is increasing or decreasing at a given point by looking at its derivative.

First, find #f'(x)# through the product rule.

#f'(x)=lnxd/dx[x^2]+x^2d/dx[lnx]#

#f'(x)=lnx*(2x)+x^2(1/x)#

#f'(x)=2xlnx+x#

Now, find #f'(1)#.

#f'(1)=2(1)ln(1)+1#

#f'(1)=2(0)+1#

#f'(1)=1#

Since #f'(1)>0#, the function is increasing at #x=1#.

We can check a graph of #f(x)#:

graph{x^2lnx [-0.353, 0.595, 1.6745, -0.221]}

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Answer 2

Cameron Alexander

To determine if the function ( f(x) = x^2 \ln x ) is increasing or decreasing at ( x = 1 ), we need to examine the sign of the derivative of the function at that point. Compute the derivative ( f'(x) ) using the product rule and evaluate ( f'(1) ). If ( f'(1) > 0 ), then the function is increasing at ( x = 1 ); if ( f'(1) < 0 ), then it is decreasing at ( x = 1 ).

Using the product rule on ( f(x) = x^2 \ln x ), we get ( f'(x) = 2x \ln x + x ). Evaluating ( f'(1) ), we find ( f'(1) = 2 \cdot 1 \cdot \ln 1 + 1 = 1 ).

Since ( f'(1) = 1 > 0 ), the function ( f(x) = x^2 \ln x ) is increasing at ( x = 1 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.