Let a function f x be continuous in an interval a , b . Let δ 0 be a very small real number. Let c ∈ a , b be such that f c - δ f c and f c + δ f c for every δ 0 . Let f α - δ - f α f α + δ - f α 0 ∀ α ∈ a , b and α ≠ c . Then

Question

Let a function f x be continuous in an interval a , b . Let δ > 0 be a very small real number. Let c ∈ a , b be such that f c - δ f c and f c + δ f c for every δ > 0 . Let f α - δ - f α f α + δ - f α 0 ∀ α ∈ a , b and α ≠ c . Then, f x has a local maximum at c and a local minimum at α, f x has a local maximum at α and a local minimum at c, f x has only one local maximum at c, f x has only one local minimum at c

MARKS App · Accepted Answer

Since, f x is continuous in a , b and f c - δ f c & f c + δ f c , threfore following scenario is possible Hence, f x must have local maximum at x = c . Also, f a − δ − f α f α + δ − f α 0 . . . 1 Case 1 : When f α - δ - f α > 0 and f α + δ - f α 0 ⇒ f α f α - δ and f α + δ f α So, f x has neither maximum nor minima at x = α . Case 2 : f α − δ − f α 0 and f α + δ − f α > 0 ⇒ f α − δ f α and f α − δ > f α Here also f x has neither maxima nor minima at x = α .

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