The function f : [ 0 , 3 ] [ 1 , 29 ] , defined by f ( x ) = 2 x 3 - 15 x 2 + 36 x + 1 , is [2012]

Question

The function $f : [0, 3] \to [1, 29]$ , defined by $f (x) = 2 x^{3} - 15 x^{2} + 36 x + 1,$ is [2012]

Smart Achievers · Accepted Answer

(2)

$Given: f (x) = 2 x^{3} - 15 x^{2} + 36 x + 1$

$\Rightarrow f' (x) = 6 x^{2} - 30 x + 36$

$= 6 (x^{2} - 5 x + 6) = 6 (x - 2) (x - 3)$

$∵$ $f' (x) > 0,$ $\forall x \in [0, 2)$ and $f' (x) < 0,$ $\forall x \in (2, 3)$

$∴$ $f (x) is increasing on [0, 2) and decreasing on (2, 3)$

$∴$ $f (x) is many-one on [0, 3]$

Also, $f (0) = 1,$ $f (2) = 29,$ $f (3) = 28$

$∴$ $Absolute minimum = 1 and Absolute maximum = 29$

$∴$ $Range of f = [1, 29] = codomain$

$∴$ $f is onto$

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