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

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

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Answer 1 · 2016-11-24T21:55:15

You have to look at the sign of the derivative of the function for #x=3#

Explanation:

#f(x) = (x-3)/sqrt(x+3) = (x-3)(x+3)^(-1/2)#

#(df)/(dx) = (x+3)^(-1/2) -1/2 (x-3)(x+3)^(-3/2)#

#(df)/(dx) |_(x=3) = 1/sqrt(6) > 0#

So the function is increasing.

On the other hand it is easy to see that #x=3# is a zero of #f(x)# and as the denominator is always positive and the numerator is negative for #x<3# and positive for #x>3# the function must be increasing.

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

1 Answer

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