Question

Which of the following function is a monotonically increasing function?

• A. $f\left(x\right)=x^2+2x$
• B. $f\left(x\right)=x^3+2x$
• C. $f\left(x\right)=x^3+2x$
• D. $f\left(x\right)=cosx$

Answer 1

The correct option is B

$f\left(x\right)=x^3+2x$

We saw that for a monotonically increasing function f’(x) $\ge$ 0. We will go through the options to find the answer.

a. f(x) = $x^2+2x$

The domain of this function is R.

f’(x) = $2x+2=2(x+1)$

2x+2 will be non negative or greater than equal to zero for $x\ge -1$. For $x<-1$, f’(x) will be negative.

So the function will be increasing only in the interval of $[-1,\infty )$ and will be decreasing in the rest of the domain. Thus the function is not monotonically increasing function

b. f(x) = $x^3+2x$

The domain for this function is also R.

f’(x) = $3x^2+2$

$3x^2+2\ge 0$ for all real numbers.

So the function will be monotonically increasing in its domain.
Derivative of sinx would give cosx and derivative of cosx would give -sinx. Since -sinx and cosx takes both positive and negative values, we can’t say both the options C and D behaves the same way throughout their domain. So, they can’t be monotonically increasing as well.

{Note - If specifically not mentioned we’ll consider all those points in the domain where the functions exists. }