Question 43: Let the function f(x) have continuous derivative on \(\mathbb{R}\) and have the graph as shown below. How many integers \(m\in (-10;10)\) are there for the function \(y=f(3 x-1)+x^{3}-3 mx\) to covariate on the interval (-2 ;first) .

\(y^{\prime}=3 f^{\prime}(3 x-1)+3 x^{2}-3 m\)

Covariant function on (-2;1) when \(y^{\prime}=3 f^{\prime}(3 x-1)+3 x^{2}-3 m \geq 0 \quad \forall x \in(-2 ; 1)\)

\(\Leftrightarrow f^{\prime}(3 x-1)+x^{2}-m \geq 0 \forall x \in(-2 ; 1) \Rightarrow m \leq g(x)=f^ {\prime}(3 x-1)+x^{2}, \forall x \in(-2 ; 1)\,\,\,

\) We have

\(f^{\prime}(3 x-1) \geq f^{\prime}(-1)=-4, x^{2} \geq 0 \Rightarrow f^{\prime}(3 x- 1)+x^{2} \geq-4\) I guess

\(\Leftrightarrow m \leq \operatorname{Min}_{(-2 ; 1)} g(x)=-4 \Rightarrow m=\{-9 ;-8 ;-7 ;-6 ;-5 ;- 4\}\)

So there are 6 integer values that satisfy the problem condition

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