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

Increasing & Decreasing Functions · Question 66 of 74

Question 66

MediumOriginal question

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

k \le 3

Correct answerYour answer

k \ge 3

Correct answerYour answer

k \ge -3

Correct answerYour answer

k \ge 9

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

$f'(x) = 3x^2 + 6x + k$ must be $\ge 0$ for every $x$ .
An upward parabola stays $\ge 0$ when its discriminant is $\le 0$ : $36 - 12k \le 0$ .
So $k \ge 3$ .

Worth noting

Reversing the inequality when solving for $k$ gives the opposite condition.

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

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