csat-suneung 2012 Q19

csat-suneung · South-Korea · csat__math-science 4 marks Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions

For a real number $m$, let $f ( m )$ be the number of intersection points of the line passing through the point $( 0,2 )$ with slope $m$ and the curve $y = x ^ { 3 } - 3 x ^ { 2 } + 1$. What is the maximum value of the real number $a$ such that the function $f ( m )$ is continuous on the interval $( - \infty , a )$? [4 points]
(1) $- 3$
(2) $- \frac { 3 } { 4 }$
(3) $\frac { 3 } { 2 }$
(4) $\frac { 15 } { 4 }$
(5) 6

For a real number $m$, let $f ( m )$ be the number of intersection points of the line passing through the point $( 0,2 )$ with slope $m$ and the curve $y = x ^ { 3 } - 3 x ^ { 2 } + 1$. What is the maximum value of the real number $a$ such that the function $f ( m )$ is continuous on the interval $( - \infty , a )$? [4 points]\\
(1) $- 3$\\
(2) $- \frac { 3 } { 4 }$\\
(3) $\frac { 3 } { 2 }$\\
(4) $\frac { 15 } { 4 }$\\
(5) 6

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