If f(x) = x^3 + bx^2 + cx + d and 0

Question

(A) Given $f(x) = x^3 + bx^2 + cx + d$.Find the derivative: $f'(x) = 3x^2 + 2bx + c$.For a quadratic expression $ax^2 + bx + c$ to be always positive,

Vedclass · Accepted Answer

$f(x)$ is an increasing function.

Vedclass · Answer

(A) Given $f(x) = x^3 + bx^2 + cx + d$.Find the derivative: $f'(x) = 3x^2 + 2bx + c$.For a quadratic expression $ax^2 + bx + c$ to be always positive,the discriminant $D$ must be less than $0$ and $a > 0$.Here,$D = (2b)^2 - 4(3)(c) = 4b^2 - 12c = 4(b^2 - 3c)$.Since $0 Thus,$D  0$.Therefore,$f'(x) > 0$ for all $x \in R$.Since $f'(x) > 0$ for all $x \in R$,$f(x)$ is a strictly increasing function on $R$.

If $f(x) = x^3 + bx^2 + cx + d$ and $0 < b^2 < c$,then on $R$,$f(x)$ is:

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