f(x) is a continuous function on mathbb{R} an | vedclass

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(A) Given the curve $y=f(x)$ passes through $(\alpha, \beta)$,we have $\beta=f(\alpha)$.Substituting this into the line equation $p\alpha+m\beta+n=0$,

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When $p+mf^{\prime}(\alpha)=0$,$px+my+n=0$ is a tangent to the curve $y=f(x)$ at $(\alpha, \beta)$.

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(A) Given the curve $y=f(x)$ passes through $(\alpha, \beta)$,we have $\beta=f(\alpha)$.Substituting this into the line equation $p\alpha+m\beta+n=0$,we see the point $(\alpha, \beta)$ lies on the line.The slope of the tangent to the curve $y=f(x)$ at $(\alpha, \beta)$ is $f^{\prime}(\alpha)$.The slope of the line $px+my+n=0$ is $-\frac{p}{m}$ (assuming $m 
eq 0$).For the line to be a tangent at $(\alpha, \beta)$,the slopes must be equal: $f^{\prime}(\alpha) = -\frac{p}{m}$,which implies $mf^{\prime}(\alpha) = -p$,or $p+mf^{\prime}(\alpha) = 0$.Therefore,when $p+mf^{\prime}(\alpha) = 0$,the line is tangent to the curve at $(\alpha, \beta)$.

$f(x)$ is a continuous function on $\mathbb{R}$ and $y=f(x)$ is a curve. If $(\alpha, \beta)$ is a point such that $\beta=f(\alpha)$ and $p\alpha+m\beta+n=0$ $(p \neq 0, m \neq 0)$,then which one of the following is true?

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