QJEEM21Madc18: JEE Main 2021 Mathematics - Class 12 JEEM21Madc18 Solution: Let f(x) be a cubic polynomial with f(1) = 10, f( 1) = 6, and…

Let f(x) = ax³ + bx² + cx + d

f'(x) = 3ax² + 2bx + c $\Rightarrow$ f''(x) = 6ax + 2b

f'(x) has local minima at x = $-$ 1, so

$\because$ f''( $-$ 1) = 0 $\Rightarrow$ $-$ 6a + 2b = 0 $\Rightarrow$ b = 3a ..... (i)

f(x) has local minima at x = 1

f'(1) = 0

$\Rightarrow$ 3a + 6a + c = 0

$\Rightarrow$ c = $-$ 9a ..... (ii)

f(1) = $-$ 10

$\Rightarrow$ $-$ 5a + d = $-$ 10 ..... (iii)

f( $-$ 1) = 6

$\Rightarrow$ 11a + d = 6 ..... (iv)

Solving Eqs. (iii) and (iv)

a = 1, d = $-$ 5

From Eqs. (i) and (ii),

b = 3, c = $-$ 9

$\therefore$ f(x) = x³ + 3x² $-$ 9x $-$ 5

So, f(3) = 27 + 27 $-$ 27 $-$ 5 = 22