Question

the graph of y = x^3 -9x^2 - 16x +1 has a slope of 5 at two points. Find the coordinates of the points

Answer 1

Find the two values of x that satisfy dy/dx = 5. Then substitute these x values into your original equation to find the corresponding y values.

Answer 2

Differentiate y = x^3 -9x^2 - 16x +1
dy/dx = 3 x^2 -18x-16
Slope of 5. Therefore,
dy/dx = 5
5= 3 x^2 -18x-16
3 x^2 -18x-21 =0
Solving for x, by factorising,
(3x-21)(x+1) = 0
Values of x= 7 and -1

To find the 2 points, solve for y by substituting x = 7 and -1

For x = 7
y = 7^3 -9(7)^2 - 16(7) +1
y=-209
(7,-209)

For x = -1
y = (-1)^3 -9(-1)^2 - 16(-1) +1
y=7
(-1,7)