NDA2017MathematicsContinuity and DifferentiabilityActual
The left-hand derivative of f( x )=[ x ] ( x ) at x = k where k is an integer and [ x ] is the greatest integer function, is
Options
- A(-1)^ k ( k -1)
- B(-1)^ k -1 ( k -1)
- C(-1)^ k k
- D(-1)^ k -1 k
Correct answer
A. (-1)^ k ( k -1)
Step-by-step solution
f(x)=[x] ( x) at x=k Left hand derivative, _ h 0 f(k)-f(k-h) h (k-integer) = _ h 0 [k] k -[k-h] (k-h) h aligned & = _ h 0 -(k-1) (k-h) h & k =0 and (k - )=(-1)^ k-1 & = _ h 0 -(k-1)-(1)^ k-1 h aligned aligned & = _ h 0 -(k-1)(-1)^ k-1 h & = (k-1)(-1)^k aligned