Answer :
Yes , if f is differentiable on an open interval then f is strictly decreasing if f'( x ) < 0 for all x in the open interval .
Given :
if f is differentiable on an open interval show f is strictly decreasing if f'(x)<0 for all x in the open interval .
Function f is differentiable on an open interval :
Example :
f ( x ) = 2x^2 + 4 is differentiable on an open interval ( 2 , 0 )
we know that
f' ( c ) = f ( b ) - f ( a )
= 2 * 0^2 + 4 - 2 * 2^2 + 4
= 4 - 8
= -4 < 0
so f' ( c ) < 0 .
Hence if f is differentiable on an open interval then f is strictly decreasing if f'( x ) < 0 .
Learn more about the interval here:
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