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

Is #f(x)=e^x/sqrt(x^2-x)# increasing or decreasing at #x=3#?

Answer 1

Calculate the derivative via the quotient rule: #f'(x)=(e^xsqrt(x^2-x)-e^x(2x-1)/(2sqrt(x^2-x)))/(x^2-x)# #=e^x(x^2-2x+1/2)/(x^2-x)^(3/2)#

If this is positive, then #f# is increasing.

#f'(3)=e^3(9-6+1/2)/(9-3)^(3/2)=7/2e^3 6^(3/2)>0#, because each of its component multiplicative parts is #>0#. So #f(x)# is increasing at #x=3#.

For a sanity check, compare the graph of the function: graph{(e^x)/sqrt(x^2-x) [-9.26, 14.7, 2, 13.99]}

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Answer 2

Elijah Abram

To determine if ( f(x) = \frac{e^x}{\sqrt{x^2 - x}} ) is increasing or decreasing at ( x = 3 ), we need to examine the derivative of ( f(x) ) at that point.

The derivative of ( f(x) ) with respect to ( x ) is:

[ f'(x) = \frac{e^x(x^2 - x)^{ - \frac{3}{2}}(x - 2)}{2} ]

Evaluate ( f'(3) ). If ( f'(3) > 0 ), then ( f(x) ) is increasing at ( x = 3 ). If ( f'(3) < 0 ), then ( f(x) ) is decreasing at ( x = 3 ).

After substitution, ( f'(3) = - \frac{e^3}{9\sqrt{2}} ).

Since ( f'(3) ) is negative, ( f(x) ) is decreasing at ( x = 3 ).

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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.