To determine

To find:

We have to find the interval on which the curve is concave downward.

Answer

The interval on which the curve is concave downward is 0,14ln⁡92.

Explanation

Calculation:

To determine concavity we calculate the second derivative of the function.

The equation of the curve is

y=2ex-e-3x

Then differentiate two times with respect to x, we have

d2yd2x=2ex-9e-3x

This curve is concave downward if d2yd2x<0.

So, to find the interval on which curve is concave downward. We have to find the value of x for which d2yd2x<0.

2ex-9e-3x<0

Or

2ex<9e-3x

Or

2e4x<9

Or

e4x<92

Taking natural logarithmic on both sides, we have

4x<ln⁡92

Or

x<14ln⁡92

So, the interval on which the curve y=2ex-e-3x is concave downward is -∞,14ln⁡92.

Conclusion:

The interval on which the curve is concave downward is 0,14ln⁡92.