Let f(x) be a polynomial of degree 5, and have extrema at x = 1 and x = –1, if , then f(2) – f(–2) is equal to: [2026]

Question

Let f(x) be a polynomial of degree 5, and have extrema at x = 1 and x = –1, if $\lim_{x \to 0} (\frac{f (x)}{x^{3}}) = - 5$ , then f(2) – f(–2) is equal to: [2026]

Smart Achievers · Accepted Answer

(4)

Degree of f(x) = 5

f(x) has extrema at x = 1 and x = –1 and $\lim_{x \to 0} (\frac{f (x)}{x^{3}}) = - 5$

$∴$ Coefficient of $x^{3}$ is –5

$∴$ f(x) must be of the form

$f (x) = a x^{5} + b x^{4} - 5 x^{3}$

$f' (x) = 5 a x^{4} + 4 b x^{3} - 15 x^{2}$

$= x^{2} (5 a x^{2} + 4 b x - 15)$

Now, x = 1 and x = –1 are roots of $f' (x) = 0$

$∴$ Sum of roots $= \frac{- 4 b}{5 a} = 0 \Rightarrow b = 0$

Products of roots $= \frac{- 15}{5 a} = - 1 \Rightarrow a = 3$

$∴ f (x) = 3 x^{4} - 5 x^{3}$

So, $f (2) - f (- 2) = [3 {(2)}^{5} - 5 {(2)}^{3}] - [3 {(- 2)}^{5} - 5 {(- 2)}^{3}]$

= 192 – 80 = 112

Solution

Q.
Let f(x) be a polynomial of degree 5, and have extrema at x = 1 and x = –1, if $\lim_{x \to 0} (\frac{f (x)}{x^{3}}) = - 5$ , then f(2) – f(–2) is equal to: [2026]

1 0

2 50

3 92

4 112

Want Us to Email you About Special Offers & Updates?