Match List-I with List-II

Consider the function $f(x) = 2x^3 − 21x^2 + 36x + 80, x∈[0, 6]$. Then

Choose the correct answer from the options given below:

(A)-(IV), (B)-(III), (C)-(I), (D)-(II)

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)