Question

Show that the function \( f(x) = x^3 + 10x + 7 \), \( x \in \mathbb{R} \), is strictly increasing.

Hint:

A function is strictly increasing if its derivative is positive everywhere.

Answer 1

A function is strictly increasing if \( f'(x)>0 \) for all \( x \in \mathbb{R} \).
\[ f(x) = x^3 + 10x + 7, \quad f'(x) = 3x^2 + 10. \] Since \( 3x^2 \geq 0 \) and \( 10>0 \), \( f'(x) = 3x^2 + 10 \geq 10>0 \) for all \( x \).
Thus, \( f(x) \) is strictly increasing.