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Many-One onto Function

If the functi...

Question

If the function $f (x) = x^{3} - k x^{2} + 5 x$ obey LMVT on the interval $[1, 2]$ at $c = \frac{7}{4}$ , then the value of $[k]$ is
( $[\cdot]$ denotes the greatest integer function)

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Solution

$f (x) = x^{3} - k x^{2} + 5 x$ , $x \in [1, 2]$
$c = \frac{7}{4}$
$f^{'} (x) = 3 x^{2} - 2 k x + 5$
$f (2) = 18 - 4 k$ , $f (1) = 6 - k$
From LMVT, $f^{'} (c) = \frac{f (2) - f (1)}{2 - 1}$
$\Rightarrow 3 {(\frac{7}{4})}^{2} - 2 k (\frac{7}{4}) + 5 = 12 - 3 k$
$\Rightarrow k = \frac{35}{8}$
$\Rightarrow [k] = 4$

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Many-One onto Function

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