Analyse the following graph of derivable of a function f(x) i.e. y = g(x) = f '(x) and answer the following question (a ≤ x ≤ b)
(i) The graph of y = f(x) will intersect x axis
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Ans.
(i)
Sol. f '(x) ≤ 0, x ∈ [a,b] so f(x) is Regular decreasing function so that where f '(x) = 0 at same point f(x) = 0. From graph f '(x) = 0 at point x = c so that f(x) = 0 at x = c.
(ii)
Sol. In this graph function is regular decreasing function. It means f '(x) doesn't change its sign. and f ' = 0 at point c, so that c is point of inflexion.
(iii)
Sol. From Rolle's theorem if f ' = f ' . Then these exist at least one point in between [a, b] for that f "(x) = 0.
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