Find the points on the curve $$ y=x^{3}+3 x^{2}-9 x+3 $$ at which the gradient is: $$ a 0 $$ b 15

Question

UpStudy AI · Accepted Answer

To find the points on the curve $$ y = x^{3} + 3x^{2} - 9x + 3 $$ where the gradient is $$0$$ and $$15$$, we need to follow these steps:

1. **Find the derivative of $$ y $$ with respect to $$ x $$ to determine the gradient function.**
   
   $$
   \frac{dy}{dx} = 3x^2 + 6x - 9
   $$

2. **Set the derivative equal to the desired gradient values and solve for $$ x $$.**

### a) Gradient = 0

$$
   3x^2 + 6x - 9 = 0
   $$
   
   Divide the entire equation by 3:

$$
   x^2 + 2x - 3 = 0
   $$
   
   Factor or use the quadratic formula:

$$
   x = \frac{-2 \pm \sqrt{4 + 12}}{2} = \frac{-2 \pm \sqrt{16}}{2} = \frac{-2 \pm 4}{2}
   $$

So, the solutions are:

$$
   x = 1 \quad \text{and} \quad x = -3
   $$

**Find the corresponding $$ y $$-values:**

- For $$ x = 1 $$:
     $$
     y = 1^3 + 3(1)^2 - 9(1) + 3 = 1 + 3 - 9 + 3 = -2
     $$
     Point: $$ (1, -2) $$

- For $$ x = -3 $$:
     $$
     y = (-3)^3 + 3(-3)^2 - 9(-3) + 3 = -27 + 27 + 27 + 3 = 30
     $$
     Point: $$ (-3, 30) $$

**Answer for part (a):**
   $$
   \boxed{\text{a) The points are }(1,\ -2)\ \text{and}\ (-3,\ 30)}
   $$

### b) Gradient = 15

$$
   3x^2 + 6x - 9 = 15
   $$
   
   Subtract 15 from both sides:

$$
   3x^2 + 6x - 24 = 0
   $$
   
   Divide the entire equation by 3:

$$
   x^2 + 2x - 8 = 0
   $$
   
   Factor or use the quadratic formula:

$$
   x = \frac{-2 \pm \sqrt{4 + 32}}{2} = \frac{-2 \pm \sqrt{36}}{2} = \frac{-2 \pm 6}{2}
   $$

So, the solutions are:

$$
   x = 2 \quad \text{and} \quad x = -4
   $$

**Find the corresponding $$ y $$-values:**

- For $$ x = 2 $$:
     $$
     y = 2^3 + 3(2)^2 - 9(2) + 3 = 8 + 12 - 18 + 3 = 5
     $$
     Point: $$ (2, 5) $$

- For $$ x = -4 $$:
     $$
     y = (-4)^3 + 3(-4)^2 - 9(-4) + 3 = -64 + 48 + 36 + 3 = 23
     $$
     Point: $$ (-4, 23) $$

**Answer for part (b):**
   $$
   \boxed{\text{b) The points are }(2,\ 5)\ \text{and}\ (-4,\ 23)}
   $$
The points where the gradient is 0 are (1, -2) and (-3, 30). The points where the gradient is 15 are (2, 5) and (-4, 23).

Find the points on the curve at which the gradient is:

b 15

Upstudy AI Solution

Answer

Solution

Extra Insights

Part (a): Gradient is 0

Part (b): Gradient is 15

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Find the points on the curve at which the gradient is: b 15