Question 49: Let the function \(y=f(x)\) have the graph of the function \(y=f'(x)\) as shown in the figure below. Function \(y=39 f(x)-8 x^{3}+45 x^{2}-276 x+1\) covariates on which of the following intervals?

We have \(y^{\prime}=39 f^{\prime}(x)-24 x^{2}+90 x-276\)

The given function is covariate \(\Leftrightarrow y^{\prime} \geq 0 \Leftrightarrow f^{\prime}(x) \geq \frac{24 x^{2}-90 x+276}{39}\)

Let (P) be the graph of the function \(y=\frac{24 x^{2}-90 x+276}{39}\)We have the graph of the function f'(x) and (P) shown in the following figure:

From the graph above, we see that the graph of the function f'(x) lies above the parabola (P) on the interval \(\left(-1 ; \frac{11}{2}\right)\)

So \(f^{\prime}(x) \geq \frac{24 x^{2}-90 x+276}{39} \Leftrightarrow x \in\left(-1 ; \frac{11}{2} \right)\)

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