Match List-I with List-II
Consider the function $f(x) = 2x^3 − 21x^2 + 36x + 80, x∈[0, 6]$. Then
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List-I |
List-II |
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(A) one of its critical points is at $x$ = |
(I) -28 |
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(B) Its absolute maximum value is |
(II) -42 |
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(C) Its absolute minimum value is |
(III) 97 |
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(D) Its second derivative at $x = 0$ is |
(IV) 6 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(IV), (B)-(III), (C)-(I), (D)-(II)
Given: $f(x)=2x^{3}-21x^{2}+36x+80,\quad x\in[0,6]$
Derivative: $f'(x)=6x^{2}-42x+36=6(x^{2}-7x+6)=6(x-1)(x-6)$
Critical points: $x=1,\;x=6$
Values:
$f(0)=80$
$f(1)=2-21+36+80=97$
$f(6)=2(216)-21(36)+36(6)+80=432-756+216+80=-28$
Second derivative: $f''(x)=12x-42 \Rightarrow f''(0)=-42$
Matching:
(A) one of its critical points is at $x=\;6\ \Rightarrow$ (IV)
(B) absolute maximum value $=97\ \Rightarrow$ (III)
(C) absolute minimum value $=-28\ \Rightarrow$ (I)
(D) $f''(0)=-42\ \Rightarrow$ (II)