The min. value of x^1 / x at x=e(x>0) is (a) e^π>π^e (b) e^π<π^e (c) e^π=π^e (d) e^π ≤π^e

Name: Problem 9, Chapter 9: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-04-14T07:06:58-08:00
Duration: 1 min 46 s
Channel: Ankit Singh
Description: The min. value of x^1 / x at x=e(x>0) is (a) e^π>π^e (b) e^π<π^e (c) e^π=π^e (d) e^π ≤π^e

step 1 So in this case we need to find minimum value of x to the power x at x equals to e. So basically

The min. value of x^1 / x at x=e(x>0) is (a) e^π>π^e (b)...

Key Concepts

Exponential and Logarithmic Functions

Exponential functions involve expressions in which a constant base is raised to a variable exponent, and they have widespread applications in growth models and differential equations. Their inverse functions, logarithms, are essential for transforming exponential relationships into linear ones, which makes analyzing and comparing such expressions much easier.

Optimization and Critical Points in Calculus

Optimization involves finding the maximum or minimum values of a function, and this process often requires taking derivatives and finding critical points. By setting the derivative equal to zero, one can determine where the function’s slopes vanish and thus identify potential extrema, which is a fundamental technique in calculus.

Functions of the Form x^(1/x)

Functions defined as x^(1/x) exhibit interesting behaviors; analyzing their derivative after taking the natural logarithm can reveal points of extremum. This approach is common when facing functions where both the base and the exponent depend on the variable, and it utilizes logarithmic differentiation to simplify the problem.

Comparing Exponential Expressions Using Logarithms

To compare expressions like a^b and b^a, logarithms are a useful tool because they convert multiplicative and exponential relationships into additive and linear ones. By taking the natural logarithm of both expressions, one can compare their magnitudes without directly computing large numbers, simplifying the analysis.

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