Find the x values at which the tangents to the graphs f(x) = 2x^2 and g(x) = x^3 have the same...

Question:

Find the {eq}x {/eq} values at which the tangents to the graphs {eq}f(x) = 2x^2 {/eq} and {eq}g(x) = x^3 {/eq} have the same slope.

Slope of a Tangent Line

The line that's tangent to a function at a point is called the tangent line. We can find the equation of a tangent line to a function by differentiating that function. That's because the value of the derivative at a point is the slope of the tangent line to that point.

Since the slope of a tangent line is equal to the derivative of a function, we need to first differentiate these two functions. This will give us the form that these slopes will take.

{eq}f'(x) = 4x\\ g'(x) = 3x^2 {/eq}

If we want to find where the tangent lines to these functions have the same slope, we thus need to set these derivatives equal. Solving the equation that arises from this will give the x coordinates where the tangent lines will have the same slope, meaning that they are parallel.

{eq}4x = 3x^2\\ 3x^2 -4x = 0\\ x(3x-4) = 0\\ x = 0\\ x = \frac{4}{3} {/eq}

The tangents to these two functions have the same slope, and are thus parallel, at the two indicated points. 