Let f be a differentiable function on R such that . Let . Then the number of times the curve meets x-axis is : [2025]

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Solution

Q.
Let f be a differentiable function on R such that $f (2) = 1, f' (2) = 4$ . Let $\lim_{x \to 0} (f {(2 + x))}^{3 / x} = e^{α}$ . Then the number of times the curve $y = 4 x^{3} - 4 x^{2} - 4 (α - 7) x - α$ meets x-axis is : [2025]

1 1

2 0

3 2

4 3

Ans.
(3)

Given, $\lim_{x \to 0} (f {(2 + x))}^{3 / x} = e^{α}$

$\Rightarrow e^{\lim_{x \to 0} \frac{3}{x} (f (2 + x) - 1)} = e^{α}$ ( $1^{\infty}$ form)

$\Rightarrow e^{\lim_{x \to 0} 3 f' (2 + x)} = e^{α}$

$\Rightarrow e^{3 f' (2)} = e^{α} \Rightarrow e^{12} = e^{α}$

So, $α = 12$

Now, $y = 4 x^{3} - 4 x^{2} - 4 (α - 7) x - α$

$= 4 x^{3} - 4 x^{2} - 20 x - 12$

$= 4 {(x + 1)}^{2} (x - 3)$

$∴$ Roots are –1, –1 and 3

So, the curve $y = 4 x^{3} - 4 x^{2} - 20 x - 12$ . The curve meets x-axis at 2 points.

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