The number of points of intersection of the t | vedclass

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(A) Let $f(x) = 5x^2 + 2x + 3 - 2\sin x$.We analyze the minimum value of $f(x) = 5(x^2 + \frac{2}{5}x) + 3 - 2\sin x = 5(x + \frac{1}{5})^2 + 3 - \fra

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$0$

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(A) Let $f(x) = 5x^2 + 2x + 3 - 2\sin x$.We analyze the minimum value of $f(x) = 5(x^2 + \frac{2}{5}x) + 3 - 2\sin x = 5(x + \frac{1}{5})^2 + 3 - \frac{1}{5} - 2\sin x = 5(x + \frac{1}{5})^2 + 2.8 - 2\sin x$.Since $-1 \le \sin x \le 1$,the term $-2\sin x$ has a minimum value of $-2$.Thus,$f(x) \ge 5(x + \frac{1}{5})^2 + 2.8 - 2 = 5(x + \frac{1}{5})^2 + 0.8$.Since $5(x + \frac{1}{5})^2 \ge 0$,it follows that $f(x) \ge 0.8 > 0$ for all real $x$.Therefore,$f(x)$ is never equal to $0$,which means the curves never intersect.The number of intersection points is $0$.

The number of points of intersection of the two curves $y = 2\sin x$ and $y = 5x^2 + 2x + 3$ is

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