Determine whether the statement is true or false. If it is...

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. $f(x)=e^{k x}$ is an increasing function if $k>0$ and a decreasing function if $k<0$.

Key Concepts

Exponential Function

An exponential function is a mathematical expression in which a constant base, often denoted by e, is raised to a variable exponent. This type of function is significant in the study of growth and decay processes and exhibits behaviors that change depending on the sign of the exponent’s coefficient.

Monotonicity

Monotonicity refers to the property of a function being either entirely non-decreasing or non-increasing over its domain. In calculus, the derivative of a function is used as a test for monotonicity; if the derivative is positive throughout an interval, the function is increasing, and if it is negative, the function is decreasing.

Transcript

00:01 In this question there is a statement and the statement is function fx equals to e raise to the power kx is an increasing function is an increasing function if k equals to k greater than zero and and a decreasing function if k less than zero.

00:30 And we have to find out whether this statement is true or false with the explanation okay so no problem here is the solution so we just take an simple standard exponent function that is a raised to the power x okay that is a raise to the power x and we know the thumb rule of the exponent if a greater than one then the function increasing okay then the function increasing and if the function increasing and if a that is the base is less than 1 then the function decreasing okay we know this thing and if the base a equals to 1 then it will be constant all over the values of x okay it will be 1 raised to the power 1 1 raise to the power 2 1 x to the 4 so if b equals a equals to 1 then it will be a constant function okay and if a the base greater than 1 it the function will be increasing if the base a less than 1 the function will be decreasing and now we will take over function that is e raise to the power k x so we can write down it okay e raise to the power kx can be written down e raise to the power k whole raise to the power x okay and now we will compare with the standard function so a is here e -raged to the power k okay so okay if k greater than 0 then okay okay okay okay if k greater than 0 then e raise to the power k greater than 1 okay because e raise to the power 0 it is so when we take k greater than 0, then e -raged to the power k greater than 1.

02:08 And if the base it is a, okay, from here we will compare you understand my point.

02:14 When the base is greater than 1, from here then the function increasing, the function increasing...

Please give Ace some feedback