To find the local minimum values of the function $f(x) = e^{2x} + e^{-x}$, we need to follow these steps:

Step 1: Find the First Derivative

Calculate the first derivative, $f'(x)$, and set it to zero to find critical points.

$f(x) = e^{2x} + e^{-x}$ Using the chain rule, we get: $f'(x) = 2e^{2x} - e^{-x}$

Step 2: Set $f'(x) = 0$

Now, we set the first derivative equal to zero to solve for $x$:

$2e^{2x} - e^{-x} = 0$

To simplify, multiply both sides by $e^x$ (to clear the negative exponent): $2e^{3x} - 1 = 0$

Now, isolate $e^{3x}$: $2e^{3x} = 1$ $e^{3x} = \frac{1}{2}$

Take the natural logarithm of both sides: $3x = \ln \left(\frac{1}{2}\right)$ $x = \frac{\ln \left(\frac{1}{2}\right)}{3} = -\frac{\ln(2)}{3}$

Thus, the critical point is: $x = -\frac{\ln(2)}{3}$

Step 3: Use the Second Derivative Test

To confirm if this point is a local minimum, we need to compute the second derivative, $f''(x)$, and evaluate it at the critical point.

$f''(x) = 4e^{2x} + e^{-x}$

Substitute $x = -\frac{\ln(2)}{3}$ into $f''(x)$: Since $f''(x)$ is positive at this point (both terms are positive for any real $x$), $x = -\frac{\ln(2)}{3}$ is a local minimum.

Step 4: Find the Minimum Value

Now, we substitute $x = -\frac{\ln(2)}{3}$ back into $f(x)$ to find the minimum value:

$f\left(-\frac{\ln(2)}{3}\right) = e^{2\left(-\frac{\ln(2)}{3}\right)} + e^{-\left(-\frac{\ln(2)}{3}\right)}$ Simplifying each term: $= e^{-\frac{2\ln(2)}{3}} + e^{\frac{\ln(2)}{3}}$ $= \left(2^{-\frac{2}{3}}\right) + \left(2^{\frac{1}{3}}\right)$ Thus, the local minimum value of $f(x)$ is $2^{-\frac{2}{3}} + 2^{\frac{1}{3}}$.

Would you like further details on any step?

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Here are some related questions to further explore:

1. How does the exponential function's behavior affect the shape of $f(x) = e^{2x} + e^{-x}$?
2. Can we generalize this method for finding minima of similar exponential functions?
3. How does the second derivative test confirm concavity for identifying minima?
4. What would be the effect of adding a constant term to $f(x)$ on its critical points?
5. Can you derive a general rule for functions of the form $f(x) = e^{ax} + e^{bx}$ to find extrema?

Tip: Remember, the second derivative test is very useful for verifying if a critical point is a minimum or maximum based on concavity.