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?Single choice

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