Given the function f(x)=x^3−6x^2+x+4, determine all coordinate points (x,y) on the graph of f such that the line tangent to fat (x,y) has a slope of 1.

Question

Given the function f(x)=x^3−6x^2+x+4, determine all coordinate points (x,y) on the graph of f such that the line tangent to fat (x,y) has a slope of 1.

Answer 1

f'(x) = 3x² - 12x + 1 = 1; 3x(x - 4) = 0; x = 0 or x = 4

If x = 0 then y = f(0) = 4, so at (0, 4) slope of tangent line is 1

If x = 4 then y = f(4) = 64 - 96 + 4 + 4 = -24, so at point (4, - 24) slope of tangent line also 1