Let f be the function given by f(x) = 3e^2x and let g be the function given by g(x) = 6x³ at what value of x do the graphs of f and g have parallel tangent lines?

Solution:

It is known that a tangent at the point to the curve has the slope same as the slope of the curve at that point.

We are given two curves:

f(x) = 3e^2x --- (1)

g(x) = 6x³ --- (2)

The slope of the curve (1) and the tangent at the point where it touches the curve is

df(x)/dx = 6e^2x --- (1)

The slope of the curve (2) and the tangent at the point where it touches the curve is

 dg(x)/dx = 18x² --- (2)

If the tangents are parallel their slopes will be same, hence

6e^2x = 18x²

Differentiating the above w.r.t x

12e^2x = 36x

Differentiating the above w.r.t x

24e^2x = 36

8 × 3e^2x = 36

8 f(x) = 36

f(x) = 9/2

y = f(x) = 9/2

Now 3e^2x = 9/2

e^2x = 3/2

Taking ln on both sides we get

ln e^2x = ln(3/2)

2x = ln(3/2)

x = 1/2ln(3/2)

x = ln√3/2

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Let f be the function given by f(x) = 3e^2x and let g be the function given by g(x) = 6x³ at what value of x do the graphs of f and g have parallel tangent lines?

Summary:

The value of x for which the tangents of the graphs of f(x) = 3e^2x and g(x) = 6x³ are parallel is ln√3/2.