Find the minimum value of the gradient of the curve

Question

y= -x^3 + 4x^2 - 7x?

Answer 1

To find gradient of curve we must fallow these 2 steps 

[1]    Find the general gradient of the curve using dy/dx

[2]    Substitute the x coordinate value of the given point
        into the expression found for dy/dx

The curve is y = - x³ + 4x² - 7x.

dy/dx = - 3x² + 4(2x) - 7

dy/dx = - 3x² + 8x - 7.

Minimum and maximum values of quadratic functions

Consider the gradient of the function with vertex .

1. If , f  has a minimum at .The minimum value is .

2. If , f  has a maximum at .The maximum value is .

The gradient of the function is  f(x) = - 3x² + 8x - 7.

Compare the above function with .

a = - 3, b = 8, and c = - 7.

Here a = - 3 < 0, so, the function doesn' t have the minimum value.

Therefore, the gradient of the given curve doesn' t have the minimum value.