For what values of
𝑥
does the function
𝑦
=
2
𝑥
3
+
9
𝑥
2
+
12
𝑥
+
6
have a gradient of zero?

Question

For what values of 
𝑥
 does the function 
𝑦
=
2
𝑥
3
+
9
𝑥
2
+
12
𝑥
+
6
  have a gradient of zero?

BlackTom AI · Accepted Answer

The derivative of y = 2x^3 + 9x^2 + 12x + 6 is y' = 6x^2 + 18x + 12.

1. Relevant knowledge points and concepts
- The gradient of a curve at a point is the derivative dy/dx. Stationary points (where the gradient is zero) occur where dy/dx = 0.
- Power rule: d/dx[x^n] = n x^(n-1).
- Solving a quadratic: factorization or the quadratic formula; simplifying by dividing by a common factor.

2. Complete reasoning process step by step
Differentiate term-by-term using the power rule:
y = 2x^3 + 9x^2 + 12x + 6
dy/dx = 3·2 x^(3-1) + 2·9 x^(2-1) + 12·1 x^(1-1) + 0
dy/dx = 6x^2 + 18x + 12

Set the gradient equal to zero to find stationary x-values:
6x^2 + 18x + 12 = 0

3. Intermediate calculation steps
Divide every term by 6 to simplify:
x^2 + 3x + 2 = 0

Factor the quadratic:
x^2 + 3x + 2 = (x + 1)(x + 2) = 0

Thus the roots are:
x + 1 = 0 => x = -1
x + 2 = 0 => x = -2

4. Why the correct answer is correct
These x-values make dy/dx exactly zero, which by definition are the points where the gradient of the original cubic is zero. The algebraic factorization shows no other real roots exist for the derivative polynomial, so there are precisely two stationary x-values: -1 and -2.

5. Why other possible choices would be incorrect
Any x-value other than -1 or -2 substituted into dy/dx = 6x^2 + 18x + 12 yields a nonzero result (for example, at x = 0, dy/dx = 12), so such values do not give a gradient of zero. The quadratic has only the two real roots found, so no other real x satisfies the condition.

6. Conclusion confirming the provided answer
The gradient is zero exactly at x = -2 and x = -1, which matches the provided answer: x = −2, x = −1.

For what values of 𝑥 does the function 𝑦 = 2 𝑥 3 + 9 𝑥 2 + 12 𝑥 + 6 have a gradient of zero?单项选择题

类似问题

Which of the following are critical points of the function \(x^3-3x^2y+3y^2\)?

Which of the following are critical points of the function [math: x3−3x2y+3y2]?

Let \(f(x) = x^{3} + 5 x^{2} - 5 x + 2.\) Which of the following is a correct classification of the critical points of \(f\)?

Let 𝑓 ( 𝑥 ) = 𝑥 3 − 5 𝑥 2 + 5 𝑥 + 2. Which of the following is a correct classification of the critical points of 𝑓 ?

Let 𝑓 ( 𝑥 ) = 𝑥 3 − 5 𝑥 2 + 5 𝑥 + 2. Which of the following is a correct classification of the critical points of 𝑓 ?

Determine all critical points for the function. f(x) = (x - 2)7

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单