Suppose that f
is an odd function for all x
. Suppose that f′(xo)
exists. Which of the following must be necessarily be equal to f′(−xo)
?
A. f′(xo)
B. 1f′(xo)
C. −f′(xo)
D. −1f′(xo)
C. -f'(xo)
Since f is an odd function, we can say that f(-x) = - f(x) for all x. So, if we differentiate both sides with respect to x, we get:
f'(-x) = -f'(x)
Now, if we substitute x = xo in the above equation, we get:
f'(-xo) = -f'(xo)
Therefore, f'(-xo) = -f'(xo), which matches option C (-f'(xo)).