A real valued function f:[4, infty) rightarro | vedclass

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(B) Given $f(x) = (x^2+x+1)^{(x^2-3x-4)}$.Taking the natural logarithm on both sides, we get $\ln(f(x)) = (x^2-3x-4) \ln(x^2+x+1)$.Differentiating wit

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monotonically increasing function

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(B) Given $f(x) = (x^2+x+1)^{(x^2-3x-4)}$.Taking the natural logarithm on both sides, we get $\ln(f(x)) = (x^2-3x-4) \ln(x^2+x+1)$.Differentiating with respect to $x$:$\frac{f'(x)}{f(x)} = (2x-3) \ln(x^2+x+1) + (x^2-3x-4) \cdot \frac{2x+1}{x^2+x+1}$.For $x \in [4, \infty)$, $x^2+x+1 > 0$ and $x^2-3x-4 = (x-4)(x+1) \ge 0$.Since $x \ge 4$, $(2x-3) > 0$ and $\ln(x^2+x+1) > 0$, and $(x^2-3x-4) \ge 0$ and $(2x+1) > 0$.Thus, $f'(x) = f(x) [ (2x-3) \ln(x^2+x+1) + \frac{(x^2-3x-4)(2x+1)}{x^2+x+1} ] > 0$ for all $x > 4$.Therefore, $f(x)$ is a monotonically increasing function.

$A$ real valued function $f:[4, \infty) \rightarrow R$ is defined as $f(x)=(x^2+x+1)^{(x^2-3x-4)}$, then $f$ is

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