"Let ƒ(x) be a polynomial of degree 3 such that ƒ(–1) = 10, ƒ(1) = –6, ƒ(x) has a critical point at x = –1 and ƒ'(x) has a critical point at x = 1. Then ƒ(x) has a local minima at x = _______."

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Accepted Answer

"Let f(x) = ax³ + bx² + cx + d

Given f(-1) = 10, f(1) = -6

$\therefore$ -a + b - c + d = 10 ....(i)

and a + b + c + d = -6 ......(ii)

adding (i) + (ii)

2(b + d) = 4

$\Rightarrow$ b + d = 2 ....(iii)

f'(x) = 3ax² + 2bx + c

Given f'(-1) = 0

$\Rightarrow$ 3a - 2b + c = 0 .....(iv)

f"(x) = 6ax + 2b

Given f"(1) = 0

$\therefore$ 6a + 2b = 0 ....(v)

$\Rightarrow$ b = -3a

adding (iv) + (v), we get

9a + c = 0 ....(vi)

$\Rightarrow$ $9\left( {{{ - b} \over 3}} \right)$ + c = 0

$\Rightarrow$ c = 3b

f(x) = ${{{ - b} \over 3}{x^3}}$ + bx² + 3bx + (2 - b)

$\Rightarrow$ f'(x) = -bx² + 2bx + 3b

= -b(x² - 2x - 3)

At maxima and minima f'(x) = 0

$\therefore$ (x² - 2x - 3) = 0

$\Rightarrow$ (x - 3) (x + 1) = 0

x = 3, -1

As a + b + c + d = -6

$\Rightarrow$ ${{{ - b} \over 3}}$ + b + 3b + 2 - b = -6

$\Rightarrow$ b = -3

$\therefore$ f'(x) = 3(x² - 2x - 3)

$\Rightarrow$ f''(x) = 3(2x - 2)

At x = 3, f''(x) = 3(2.3 - 2) = 12 > 0

$\therefore$ Minima at x = 3."

Let ƒ(x) be a polynomial of degree 3 such that ƒ(–1) = 10, ƒ(1) = –6, ƒ(x) has a critical point at x = –1 and ƒ'(x) has a critical point at x = 1. Then ƒ(x) has a local minima at x = _______.

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Let ƒ(x) be a polynomial of degree 3 such that ƒ(–1) = 10, ƒ(1) = –6, ƒ(x) has a critical point at x = –1 and ƒ'(x) has a critical point at x = 1. Then ƒ(x) has a local minima at x = _______.

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