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Implicit Differentiation

Let f:R→ R ...

Question

Let $f : R \to R$ & $^{'} k^{'}$ be largest natural number for which $f (x) = x^{3} + 2 k x^{2} + (k^{2} + 12) x - 12$ is a bijective function, then $\frac{f (1) + f^{- 1} (49)}{10}$ is equal to

1

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2

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5

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4

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Solution

The correct option is B $5$
Derivative of $f (x)$
$f^{'} (x) = 3 x^{2} + 4 k x + (k^{2} + 12) \geq 0 \forall x ϵ R$

Above equation is in the form of quadratic equation determinant should be negative or equal to zero.

$D \leq 0 \Rightarrow 16 k^{2} - 12 (k^{2} + 12) \leq 0$

$\Rightarrow$ $k^{2} - 36 \leq 0$
$\Rightarrow$ $k ϵ [- 6, 6]$
$\Rightarrow$ $k = 6$
$\Rightarrow$ $f (1) = 1 + 12 + 48 - 12 = 49$ & $f^{- 1} (49) = 1$
$\Rightarrow$ $f (1) + f^{- 1} (49) = 50$
$\Rightarrow$ $\frac{f (1) + f^{- 1} (49)}{10} = 5$

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