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Question
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.
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Solution
f(x) = 4x3 – 18x2 + 27x – 7
f′(x) = 12x2 – 36x + 27
= 3(4x2 – 12x + 9)
= 3(2x – 3)2
f'(x) = 0
⇒ x = `3/2` .....(critical point)
Since f′(x) > 0 for all x < `3/2` and for all x > `3/2`
Hence x = `3/2` is a point of inflexion
i.e., neither a point of maxima nor a point of minima.
x = `3/2` is the only critical point, and f has neither maxima nor minima.
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