Let f(x) be a cubic polynomial with f(1) = $-$10, f($-$1) = 6, and has a local minima at x = 1, and f'(x) has a local minima at x = $-$1. Then f(3) is equal to ____________.

<p>Let f(x) = ax³ + bx² + cx + d</p> <p>f'(x) = 3ax² + 2bx + c $\Rightarrow$ f''(x) = 6ax + 2b</p> <p>f'(x) has local minima at x = $-$1, so</p> <p>$\because$ f''($-$1) = 0 $\Rightarrow$ $-$6a + 2b = 0 $\Rightarrow$ b = 3a ..... (i)</p> <p>f(x) has local minima at x = 1</p> <p>f'(1) = 0</p> <p>$\Rightarrow$ 3a + 6a + c = 0</p> <p>$\Rightarrow$ c = $-$9a ..... (ii)</p> <p>f(1) = $-$10</p> <p>$\Rightarrow$ $-$5a + d = $-$10 ..... (iii)</p> <p>f($-$1) = 6</p> <p>$\Rightarrow$ 11a + d = 6 ..... (iv)</p> <p>Solving Eqs. (iii) and (iv)</p> <p>a = 1, d = $-$5</p> <p>From Eqs. (i) and (ii),</p> <p>b = 3, c = $-$9</p> <p>$\therefore$ f(x) = x³ + 3x² $-$ 9x $-$ 5</p> <p>So, f(3) = 27 + 27 $-$ 27 $-$ 5 = 22</p>