10. A curve has the equation y = ax^3 + x + c, where a and c are constants. The gradient of the curve at the
point (1, 2) is -2.
(i) Find the value of a and of c.
(ii) Hence, find the coordinates of the other point
on the curve where the gradient is -2.

Part I)

Giving the equation:

Part II)

(-1, 2)

Step-by-step explanation:

We have the equation:

Where a and c are constants.

We are given gradient/slope of the curve at the point (1, 2) is -2.

Part I)

We know that the gradient of the curve exactly at the point (1, 2) is -2.

So, we will differentiate our equation. We have:

Differentiate both sides with respect to x:

Expand the right:

Since a and c are simply constants, we can move them outside:

Differentiate. Note that the derivative of a constant is simply 0.

We know that at the point (1, 2), the gradient/slope is -2.

Hence, when x is 1 and when y is 2, y’ is -2.

So (we don’t have a y so we can ignore it):

Solve for a. Simplify:

Subtract 1 from both sides yield:

So, it follows that:

Therefore, our derivative will be:

Now, we can go back to our original equation:

Since we know that a is -1:

Recall that we know that a point on our curve is (1, 2).

So, when x is 1, y is 2. By substitution:

Solve for c. Simplify:

Then it follows that:

Hence, our entire equation is:

Part II)

Our derivative is:

If the gradient is -2, y’ is -2. Hence:

Solve for x. Subtracting 1 from both sides yields:

So:

Then it follows that:

Since we already have the point (1, 2) where x is 1, our other point is where x is -1.

Using our equation:

We can see that our second point is:

Simplify:

So, our second point where the gradient is -2 is at (-1, 2).

8(-10x+3.5y-7)

8×-10x + 8×3.5y + 8×-7

-80x + 28y - 56

therefore

-80x+28y-56 is the correct answer