Let f be a polynomial function such that f \left(\right. f \left(\right. x \left.\right) \left.\right) - 8 x^{m} = \frac{80}{f ' ' \left(\right. x \left.\right)} – 64x2 + 100 ∀x ∈ R, m ∈ N and f \textrm{ } \left(\right. \left|\right. x \left|\right. \left.\right) is derivable for all x ∈ R. Find the value of \left[\right. \frac{f \left(\right. 3 \left.\right)}{m} \left]\right.. [Note: [ k] denotes greatest integer function less than or equal to k.]

Engineering

Mathematics

Successive Differentiation

Question

Let f be a polynomial function such that $f (f (x)) - 8 x^{m} = \frac{80}{f'' (x)}$ – 64x² + 100 ∀x ∈ R, m ∈ N and $f (| x |)$ is derivable for all x ∈ R. Find the value of $[\frac{f (3)}{m}]$ .

[Note: [ k] denotes greatest integer function less than or equal to k.]

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f (x) must be a quadratic of the form

f (x) = ax² + b.

Now, substituting f (x) in the given relation

a(ax² + b)² + b – 8xm = $\frac{80}{2 a} - 64 x^{2} + 100$

a³x⁴ + 2a²bx² + ab² + b – 8xm = $\frac{40}{a} - 64 x^{2} + 100$

m must be equal to 4 and a³ – 8 = 0

⇒ a = 2

2a²b = – 64 ⇒ b = – 8

$\underset{120}{\underset{⏟}{{ab}^{2} + b}} = \underset{120}{\underset{⏟}{\frac{40}{a} + 100}}$

$∴$ f (x) = 2x² – 8

f (3) = 10 $\Rightarrow [\frac{f (3)}{m}] = [\frac{10}{4}] = 2$