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प्रश्न
Find the point of local maximum or local minimum, if any, of the following function, using the first derivative test. Also, find the local maximum or local minimum value, as the case may be:
f(x) = x3(2x \[-\] 1)3.
उत्तर
We have, f(x) = x3(2x − 1)3
Differentiate w..r.t x, we get,
f'(x) = 3x2(2x − 1)3 + 3x3(2x − 1)2·2
= 3x2(2x − 1)2(2x − 1 + 2x)
= 3x2(4x − 1)
For the point of local maxima and minima,
f'(x) = 0
= 3x2(4x − 1) = 0
= x = 0, `1/4`
At x = 1/4 f'(x) changes from −ve to + ve
Since, x = `1/4` is a point of minima
Hence, local min value f`(1/4) = -1/512`
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