e^x>10 x is satisfied (a) for all x (b) for all x>3 (c) for x only if x ≥20 (d) for all x>18

Name: Problem 36, Chapter 24: Comprehensive Mathematics for JEE Advanced
Uploaded: 2021-12-02T02:02:20-08:00
Duration: 2 min 3 s
Channel: Aman Gupta
Description: e^x>10 x is satisfied (a) for all x (b) for all x>3 (c) for x only if x ≥20 (d) for all x>18

step 1 in this problem let us take a function f x which is equals to e power of x minus 10x which makes

E^x>10 x is satisfied (a) for all x (b) for all x>3 (c) for...

Key Concepts

Transcendental Equations

Transcendental equations involve functions that are not algebraic, such as those including exponential, logarithmic, or trigonometric terms. The equation e^x = 10x is an example, where finding an exact algebraic solution is not straightforward. Such equations typically require numerical methods or graphical analysis to estimate solutions and understand the conditions under which one function exceeds another.

Exponential Function Behavior

Exponential functions, such as e^x, grow rapidly as x increases. They are defined for all real numbers and have continuous, smooth curves. Their growth rate is proportional to their current value, meaning that as x increases, the rate of increase also accelerates. This makes exponential functions eventually dominate over linear functions for sufficiently large values of x.

Linear Function Behavior

Linear functions, like 10x, increase at a constant rate. They produce a straight line when graphed, meaning that the increase in function value is directly proportional to the increase in x. Compared to exponential functions, linear functions do not exhibit accelerating growth, which is critical when comparing their behavior for large values of x.

Comparison of Growth Rates

Comparing the growth rates of different types of functions, particularly exponential versus linear, is essential in understanding inequalities like e^x > 10x. This concept involves analyzing how rapidly each function increases as x becomes larger. Due to the accelerating growth of exponential functions, there exists a critical value beyond which the exponential function will always exceed a linear function, even if the linear function has a relatively high constant multiplier.

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