NDA 2025 Maths · Mathematics · Application of Derivatives · hard

F(x)=e^x is increasing for all real x. 2. f(x)=x^3 is increasing for all real x.

Single correct option

Consider the following statements:
1.

f(x)=e^x

is increasing for all real

x

.
2.

f(x)=x^3

is increasing for all real

x

.
Which of the statements given above is/are correct?

Show answer and solution

Answer: (C) Both $1$ and $2$

f'(x)=e^x>0

for all

x

, so statement 1 holds. For

x^3

f'(x)=3x^2\ge0

and vanishes only at

x=0

, an isolated point, so the function is still strictly increasing on

\mathbb R

. Statement 2 also holds. Zero derivative at one point does not stop a function from increasing.

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