f(x) = kx^3 - 9x^2 + 9x + 3 is increasing on R. Which condition must k satisfy?

Increasing & Decreasing Functions · Question 74 of 74

Question 74

HardOriginal question

$f(x) = kx^3 - 9x^2 + 9x + 3$ is increasing on $\mathbb{R}$ . Which condition must $k$ satisfy?

k > 0

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k \le 3

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k \ge 9

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k \ge 3

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Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

$f'(x) = 3kx^2 - 18x + 9$ must be $\ge 0$ for every $x$ : the parabola must open upwards ( $k > 0$ ) with discriminant $\le 0$ .
$324 - 108k \le 0$ gives $k \ge 3$ , which already makes $k > 0$ .

Worth noting

$k > 0$ only makes the parabola open upwards; it can still dip below 0, as for $k = 1$ , where $3x^2 - 18x + 9$ is negative at $x = 3$ .

Ten questions from Increasing & Decreasing Functions, at your own pace.

Ten questions from Increasing & Decreasing Functions, at your own pace.