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$If a < (28)^{\frac{1}{3}} - 3 < b, then (a, b) is$

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a

128,127

b

128,129

c

27,28

d

127,126

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detailed solution

Correct option is A

Let f(x)=x13,x∈[27,28]Differentiating with respect to x on both sides⇒f'x=13x23 By Lagrange's Mean value theorem: f(b)−f(a)b−a=f′(c), where c∈(27,28)⇒(28)13−(27)1328−27=f′(c)⇒(28)23−3=13c23 Now, 271c23>12823⇒127>13.c23>13(28)23=(28)133128>128∴128<13(c)23<127⇒128<(28)13−3<127

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Consider the function $f (x) = 8 x^{2} - 7 x + 5$ on the interval $[- 6, 6] .$ the value of c satisfying the conclusion of Lagrange’s mean value theorem is ______________.

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