Question

Can you please help

Answer 1

Lets first find the derivative of f:

`(d)/(dx)(e^(2x)+e^(-x))=e^(-x)(2e^(3x)-1)`

Now lets set the derivative equal to zero to find the minimum value:

`e^(-x)(2e^(3x)-1)=0`
`(2e^(3x)-1)=0`
`x=(-ln(2))/(3)`

And replacing that value of x, we have:

`e^{2((-\ln(2))/(3))}+e^{-((-\ln(2))/(3))}`
`e^{(-2\ln (2))/(3)}+e^{((\ln (2))/(3))}`
`\frac{3\sqrt[3]{2}}{2}`

The local minimum of f is at:

`((-ln(2))/(3),\frac{3\sqrt[3]{2}}{2})`