How do I find a common tangent line between two functions?

Question

Find the point where the graphs of f(x)=x^3-2x and g(x)=0.5x^2-1.5 are tangent to each other; that is have a common tangent line.

Use you knowledge of the derivative to solve this problem. write out a complete solution with explanation.

I can see it in the graph but am unsure how to come up with the solution and what work is required. I know f'(x)=3x^2-2 and g'(x)=2(0.5x)

Answer 1

a tangent will have the same derivative. so since it's a common tangent the derivatives will be equal
3x^2 -2 = x { 2(0.5x) = x}
3x^2 - x - 2 =0
(3x+2)(x-1)=0

x = -2/3 or x = 1

Answer 2

Common Tangent

Answer 3

f'(x)=3x^2-2 and g'(x)=2(0.5x)

As the tangent is common, the gradients are also common.

3x^2-2 = 2(0.5x)
3x^2 - x - 2 = 0
(3x+2)(x-1) = 0
x = -2/3 or 1

f(-2/3) = -(8/27) + 4/3 = 28/27
f(1) = 1

Find the equations with these 2 points (HINT: Using y=mx+c) and you will get 2 equations.