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Question
f(x) and f’(x) are differentiable at x = c. A sufficient condition for f(c) to be an extremum of f(x) is that f’(x) changes sign as x passes through c
f(x) and f’(x) are differentiable at x = c. A sufficient condition for f(c) to be an extremum of f(x) is that f’(x) changes sign as x passes through c
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Solution
The correct option is A True
We know that f(x) will always have an extremum if f’(x) changes its sign in the neighborhood of a point. If f’(c+h) < 0 & f’(c-h) > 0 then we will have a maximum at x = c given that the function is differentiable at x =c. And if f’(c+h) > 0 & f’(c-h) < 0 then we will have a minimum at x = c.
So, if we know that f’(x) is changing its sign in the neighborhood of some point “c” and also f(x) is differentiable at x = c then we can say that f(x) has an extremum at x = c.
[h is tending to zero from right side]
True
We know that f(x) will always have an extremum if f’(x) changes its sign in the neighborhood of a point. If f’(c+h) < 0 & f’(c-h) > 0 then we will have a maximum at x = c given that the function is differentiable at x =c. And if f’(c+h) > 0 & f’(c-h) < 0 then we will have a minimum at x = c.
So, if we know that f’(x) is changing its sign in the neighborhood of some point “c” and also f(x) is differentiable at x = c then we can say that f(x) has an extremum at x = c.
[h is tending to zero from right side]
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