Let fx-x6-2x4-x3-2x-3-x- Then the natural number n for which lim x- 1xnf1-fxx-1-44 is

Lim_x→ 1x^nf(1) f(x)x 1=44 By L.H. Rule ⇒lim_x→ 1 ( nx^n 1f(1) f'(x) )=44 ⇒ n.f(1) f'(1) = 44 ⋯(i) f(x)=x^6+2x^4+x^3+2x+3 ⇒ f(1)=9,f'(1)=19 Using above va ...

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Standard XII

Mathematics

Right Hand Derivative

Let fx=x6+2x4...

Question

Let $f (x) = x^{6} + 2 x^{4} + x^{3} + 2 x + 3, x \in R .$ Then the natural number $n$ for which $lim x \to 1 \frac{x^{n} f (1) - f (x)}{x - 1} = 44$ is

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Solution

$lim x \to 1 \frac{x^{n} f (1) - f (x)}{x - 1} = 44$

By L.H. Rule
$\Rightarrow lim x \to 1 (n x^{n - 1} f (1) - f^{'} (x)) = 44$
$\Rightarrow n . f (1) - f^{'} (1) = 44 \dots (i)$
$f (x) = x^{6} + 2 x^{4} + x^{3} + 2 x + 3$
$\Rightarrow f (1) = 9, f^{'} (1) = 19$
Using above values in $(i),$ we get
$n (9) - 19 = 44$
$∴ n = 7$

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Let f(x)=x6+2x4+x3+2x+3,x∈R. Then the natural number n for which limx→1xnf(1)−f(x)x−1=44 is

limx→1xnf(1)−f(x)x−1=44 By L.H. Rule ⇒limx→1(nxn−1f(1)−f′(x))=44 ⇒n.f(1)−f′(1)=44 ⋯(i) f(x)=x6+2x4+x3+2x+3 ⇒f(1)=9,f′(1)=19 Using above values in (i), we get n(9)−19=44 ∴n=7

Let $f (x) = x^{6} + 2 x^{4} + x^{3} + 2 x + 3, x \in R .$ Then the natural number $n$ for which $lim x \to 1 \frac{x^{n} f (1) - f (x)}{x - 1} = 44$ is