Use the given derivative to find all critical points of f...

Use the given derivative to find all critical points of $f$ and at each critical point determine whether a relative maximum, relative minimum, or neither occurs. Assume in each case that $f$ is continuous everywhere.
$$
f^{\prime}(x)=e^{2 x}-5 e^{x}+6
$$

Key Concepts

Critical Points

Critical points of a function are points in its domain where the derivative either equals zero or does not exist. These points are important because they are potential locations for local maxima, local minima, or inflection points, where the function's behavior might change.

First Derivative Test

The first derivative test is a method used to classify critical points by examining the sign of the derivative before and after these points. A change from positive to negative indicates a relative maximum, while a change from negative to positive indicates a relative minimum. If there is no sign change, the point is neither a maximum nor a minimum.

Substitution Method for Solving Equations

Substitution is a useful technique for simplifying complex equations, particularly when dealing with repeated exponential or other composite functions. By replacing a complex expression with a simpler variable, the equation can often be transformed into a more manageable form, such as a polynomial equation, which can then be solved more easily.

Continuity of Functions

Continuity is a fundamental property that ensures a function has no breaks, jumps, or holes in its domain. When a function is continuous everywhere, techniques such as the first derivative test can be applied confidently, knowing that the behavior of the function is well-behaved across its entire domain.

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