How do you find the value of k where the 2 function continuous #g(x)= |x^3|# for #-oo<x<k# and #x^2# for #k<x<oo#?

1 Answer
sente
Oct 13, 2016

#k in {-1, 0, 1}#

Explanation:

For #g(x)={(|x^3| if x<=k), (x^2 if x>=k):}# to be continuous, we must have the piecewise components agree at the value at which they meet, i.e. we must have #|k^3| = k^2#.

We'll consider two cases:

Case 1: #k >= 0#

#=> |k^3| = k^3#

#=> k^3 - k^2 = 0#

#=> k^2(k-1) = 0#

#=> k = 0 or k = 1#

Case 2: #k < 0#

#=> |k^3| = -k^3#

#=> -k^3 - k^2 = 0#

#=> -k^2(k+1) = 0#

#=> k = -1#

Together, we get three possible values for #k# which will result in #g# being continuous: #k in {-1, 0, 1}#

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