For what values of K is the function f(x) = K | vedclass

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(C) Given $f(x) = Kx^3 + 9x^2 + 9x + 3$. For $f(x)$ to be an increasing function on $R$,we must have $f'(x) \geq 0$ for all $x \in R$. First,find the

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$K \leq 3$

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(C) Given $f(x) = Kx^3 + 9x^2 + 9x + 3$. For $f(x)$ to be an increasing function on $R$,we must have $f'(x) \geq 0$ for all $x \in R$. First,find the derivative: $f'(x) = 3Kx^2 + 18x + 9$. We require $3Kx^2 + 18x + 9 \geq 0$ for all $x \in R$. Dividing by $3$,we get $Kx^2 + 6x + 3 \geq 0$. For a quadratic $ax^2 + bx + c \geq 0$ to hold for all $x \in R$,we must have $a > 0$ and the discriminant $D = b^2 - 4ac \leq 0$. Here,$a = K$,$b = 6$,and $c = 3$. Condition $1$: $K > 0$. Condition $2$: $D = 6^2 - 4(K)(3) \leq 0$. $36 - 12K \leq 0$. $36 \leq 12K$. $K \geq 3$. Combining $K > 0$ and $K \geq 3$,we get $K \geq 3$. Thus,the function is increasing for $K \geq 3$.

For what values of $K$ is the function $f(x) = Kx^3 + 9x^2 + 9x + 3$ an increasing function on $R$?

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