here is the question:
determine the minimum gradient of the curve y=2^3 - 9x^2 + 5
when they say the minimum gradient, does that mean the minimum value of d^2y/dx^2?
how do i do this??
Bailey, Andrea, Danny, Riley, anabelle, holly, LILLY, zachary, I'm stumped -- or whoever!
Please post under one name. There's no reason to post under several.
They want the minimum value of dy/dx. That would be where the second derivative is zero, but you have to test that it isn't a maximum.
You seem to have mistyped the function, which probably should be
y(x) = 2x^3 -9x^2 +5
dy/dx = 6x^2 -18x
d^2y/dx^2 = 12x -18
That = 0 when x = 3/2
Since the third derivative is positive, x = 3/2 is a minimum gradient location
To determine the minimum gradient of a curve, you need to find the minimum value of the derivative of the curve. In this case, since you mentioned the second derivative, you are correct that finding the minimum gradient refers to finding the minimum value of d^2y/dx^2.
To find the second derivative, follow these steps:
1. Start by finding the first derivative, dy/dx, of the given curve. To do this, differentiate each term of the equation with respect to x, using the power rule for differentiation.
For the equation y = 2^3 - 9x^2 + 5:
- The derivative of 2^3 is 0, as it is a constant.
- The derivative of -9x^2 is -18x.
- The derivative of 5 is 0, as it is a constant.
Therefore, dy/dx = -18x.
2. Once you have the first derivative, find the second derivative, d^2y/dx^2, by differentiating the first derivative with respect to x.
In this case, the first derivative is dy/dx = -18x.
Differentiating dy/dx = -18x with respect to x, you get d^2y/dx^2 = -18.
Now, the second derivative d^2y/dx^2 is a constant value of -18. Since it is the only value it can take, it means that the gradient of the curve is always decreasing (negative) with a constant slope of -18. Therefore, the minimum gradient of the curve y = 2^3 - 9x^2 + 5 is -18.