Increasing Function and its Derivative

Very Hard Algebra Functions

Let the function $$ f $$ be defined by the equation:

$$ f(x) = ax^3 + bx^2 + cx + d $$

Given that the function is increasing for all real values of $$ x $$, what can be inferred about the coefficients $$ a $$, $$ b $$, $$ c $$, and $$ d $$?

If $$ a = 2 $$, $$ b = 0 $$, $$ c = 3 $$, and $$ d = -5 $$, what is the expression for the derivative of the function, and what does it tell us about the function's behavior?

$$ f'(x) = 6x^2 + 3 $$, which is always positive, indicating that $$ f $$ is always increasing.

$$ f'(x) = 6x^2 - 3 $$, which has both positive and negative intervals, indicating that $$ f $$ is not always increasing.

$$ f'(x) = 6x^2 + 3 $$, which is not a valid form as it suggests the function can be constant.

$$ f'(x) = 6x^2 - 3 $$, which leads to limited intervals of increase for $$ f $$.

Hint

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