The minimum value of the function y=2x3-21x2+36x-20 is:

Question

The minimum value of the function $y = 2 x^{3} - 21 x^{2} + 36 x - 20$ is:

Smart Achievers · Accepted Answer

(3)

Ans. - 128

Explanation:

$y = f (x) = 2 x^{3} - 21 x^{2} + 36 x - 20$

$f' (x) = 6 x^{2} - 42 x + 36$

$f'' (x) = 12 x - 42$

For maxima/minima point, $f (x) = 0$

$6 x^{2} - 42 x + 36 = 0$

$\Rightarrow$ $x^{2} - 7 x + 6 = 0$

$\Rightarrow$ $(x - 6) (x - 1) = 0$ $\Rightarrow$ $x = 6, 1$

$∴$ $f'' (6) = 72 - 42 = 30 > 0$

$f'' (1) = 12 - 42 = - 30 < 0$

Hence, the minimum point is 6.

The minimum value is

$f (6) = 2 {(6)}^{3} - 21 {(6)}^{2} + 36 (6) - 20$

$= 432 - 756 + 216 - 20$ $= - 128 .$

$∴$ Minimum value = $- 128 .$

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