Question

The function f : [0, 3] → [1, 29], defined by f(x) = 2x³ - 15x² + 36x + 1 is: a) One-one & Onto b) Onto but not One-one c) One-one but not Onto d) Neither One-one nor Onto (Steps are mandatory) ( JEE - Adv. 2012 ) [ +3 , -1 ]

Answer 1

f(x) = 2x3 - 15x2 + 36x + 1

f'(x) = 6x2 - 30x + 36

= − 6(x2 - 5x + 6)

= 6(x - 2)(x - 3)

f(x) is increasing in [0, 2] and decreasing in [2, 3]. f (x) is many one.

f(0) = 1 f(2)

= 29 f(3)

= 28 Range is [1, 29].

Hence, f (x) is many-one-onto.

Answer 2

ANSWER.

Option [ b ] is correct answer.

EXPLANATION.

$\sf : \implies \: { \underline{conditions \: of \: one - one \: and \: onto}} \\ \\ \sf : \implies \: \: one - one \: we \: must \: check \: f {}^{' } (x) > 0 \: \: or \: f {}^{' } (x) < 0 \: in \: given \: domain \\ \\ \sf : \implies \: in \: onto \: function \: range = codomain$

$\sf : \implies \: f \ratio \: (0 , 3) \to \: (1 , 29) \\ \\ \sf : \implies \: f(x) = 2 {x}^{3} - 15 {x}^{2} + 36x + 1 \\ \\ \sf : \implies \: we \: can \: differentiate \: the \: equation \\ \\ \sf : \implies \: f {}^{ ' } (x) = 6 {x}^{2} - 30x + 36 \\ \\ \sf : \implies f {}^{ ' } (x) = 6( {x}^{2} - 5x + 6) \\ \\ \sf : \implies \: 6( {x}^{2} - 3x - 2x + 6)$

$\sf : \implies \: f {}^{ ' } (x) = 6(x - 2)( x - 3) \\ \\ \sf : \implies \: we \: can \: put \: the \: value \: in \:wavy \: curve \: method \\ \\ \sf : \implies \: zeroes \: of \: equation \: are =( 0,2,3) \\ \\ \sf: \implies \: the \: value \: given \: is \\ \\ \sf : \implies \:for \: given \: domain \: (0,3) \: \\ \\ \sf : \implies \: f(x) \: is \: increasing \: as \: well \: as \: decreasing \: \implies \: many \: one \:$

$\sf : \implies \: now \: put \: the \: value \: f {}^{'}(x) = 0 \\ \\ \sf : \implies \: 2(0 {}^{3} ) - 15( {0}^{2} ) + 36(0) + 1 \\ \\ \sf : \implies \: f(x) =1 \\ \\ \sf : \implies \: put \: f(2) \\ \\ \sf : \implies \: 2(2 {}^{3}) - 15(2 {}^{2} ) + 36(2) + 1 \\ \\ \sf : \implies \: 16 - 60 + 72 + 1 = 29 \\ \\ \sf : \implies \: f(2) = 29$

$\sf : \implies \: f(3) \\ \\ \sf : \implies \: 2(3 {}^{3}) - 15(3 {}^{2} ) + 36(3) + 1 \\ \\ \sf : \implies \: 54 - 135 + 108 + 1 \\ \\ \sf : \implies \: f(3) = 28$

$\sf : \implies \: range \: = (1 , 29) \\ \\ \sf : \implies \: \green{{ \underline{function \: is \: onto \: but \: not \: one - one}}}$