If the function f(x) = x^3 - 9kx^2 + 27x + 30 is increasing on R,… - Vidyadip

MCQ

If the function $f(x) = x^3 - 9kx^2 + 27x + 30$ is increasing on $R,$ then:

✓
$-1\leq\text{k}\leq1$
B
$k < -1$ or $k > 1$
C
$0 < k < 1$
D
$-1 < k < 0$

✓

Answer

Correct option: A.

$-1\leq\text{k}\leq1$

$f(x) = x^3 - 9kx^2 + 27x + 30$
$\Rightarrow f'(x) = 3x^3 - 18kx + 27$
$\Rightarrow 3(x^2 - 6kx + 9)$
Function is always increasing on $R.$
$3(x^2 - 6kx + 9) > 0$
$x^2 - 6kx + 9 > 0$
In $ax^2 + bx + c = 0$ if $a > 0$
$\Rightarrow b^2 - 4ac < 0$
$36k^2 - 36 < 0$
$k^2 - 1 < 0$
$(k + 1)(k - 1) < 0$
$\Rightarrow -1 < k < 1$

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