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Question
A function f(x) will have a local minimum at x = c if [ h is positive and tends to zero]
A function f(x) will have a local minimum at x = c if [ h is positive and tends to zero]
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Solution
The correct option is C f(c-h) > f(c) & f(c) < f(c+h)
From the definition of local minimum, we know that it is a point which has lower output compared to its neighborhood .In other words f(x) will have a local minimum at x = c if f(c) < f(c-h) and f(c) < f(c+h) where h is an infinitesimal positive number. So option C is correct. Option A is the condition for monotonically increasing function.
And option B is the condition for local maximum.
f(c-h) > f(c) & f(c) < f(c+h)
From the definition of local minimum, we know that it is a point which has lower output compared to its neighborhood .In other words f(x) will have a local minimum at x = c if f(c) < f(c-h) and f(c) < f(c+h) where h is an infinitesimal positive number. So option C is correct. Option A is the condition for monotonically increasing function.
And option B is the condition for local maximum.
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