Curve with minimum point

A question is this type if and only if it involves finding the exact coordinates of a minimum point on a curve (typically involving ln(x) or exponential functions) and then finding an area using integration by parts.

6 questions · Standard +0.3

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CAIE P3 2004 June Q10
11 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{2718ebbb-29e3-46f7-8d8d-ec7d526483f8-3_458_920_1144_609} The diagram shows the curve \(y = \frac { \ln x } { x ^ { 2 } }\) and its maximum point \(M\). The curve cuts the \(x\)-axis at \(A\).
  1. Write down the \(x\)-coordinate of \(A\).
  2. Find the exact coordinates of \(M\).
  3. Use integration by parts to find the exact area of the shaded region enclosed by the curve, the \(x\)-axis and the line \(x = \mathrm { e }\).
CAIE P3 2006 June Q8
9 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{88f67166-7b44-4b04-b323-f43827531495-3_558_1047_950_550} The diagram shows a sketch of the curve \(y = x ^ { \frac { 1 } { 2 } } \ln x\) and its minimum point \(M\). The curve cuts the \(x\)-axis at the point \(( 1,0 )\).
  1. Find the exact value of the \(x\)-coordinate of \(M\).
  2. Use integration by parts to find the area of the shaded region enclosed by the curve, the \(x\)-axis and the line \(x = 4\). Give your answer correct to 2 decimal places.
  3. Express \(\frac { 10 } { ( 2 - x ) \left( 1 + x ^ { 2 } \right) }\) in partial fractions.
  4. Hence, given that \(| x | < 1\), obtain the expansion of \(\frac { 10 } { ( 2 - x ) \left( 1 + x ^ { 2 } \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2010 November Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{bbc19395-6f88-4a7c-b5d4-59ced9ccdcf2-4_597_895_258_625} The diagram shows the curve \(y = x ^ { 3 } \ln x\) and its minimum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. Find the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 2\).
CAIE P3 2010 November Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{822f851a-7fae-43b8-9ebc-94588f01e51c-4_597_895_258_625} The diagram shows the curve \(y = x ^ { 3 } \ln x\) and its minimum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. Find the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 2\).
CAIE P3 2011 November Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{f421f03c-57c9-4feb-91b9-a7f9b12f96ce-3_666_956_1231_593} The diagram shows the curve \(y = x ^ { 2 } \ln x\) and its minimum point \(M\).
  1. Find the exact values of the coordinates of \(M\).
  2. Find the exact value of the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = \mathrm { e }\).
CAIE P3 2011 November Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{9e129863-5994-4e17-81f8-e139515998d1-3_666_956_1231_593} The diagram shows the curve \(y = x ^ { 2 } \ln x\) and its minimum point \(M\).
  1. Find the exact values of the coordinates of \(M\).
  2. Find the exact value of the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = \mathrm { e }\).