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Bài tập CALCULUS 56
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Created by T. Madas
Question 197
(****)
The cubic curve with equation
y = ax3 + bx 2 + cx + d ,
where a , b , c are non zero constants and d is a constant, has one local maximum
and one local minimum.
Show clearly that
b 2 > 3ac
SYN-B , proof
Created by T. Madas
Created by T. Madas
Question 198
(****)
y
L1
y = 2 x2 − x + 3
Q
P
R
O
x
L2
The figure above shows the curve C with equation
y = 2 x2 − x + 3 .
C crosses the y axis at the point P . The normal to C at P is the straight line L1 .
a) Find an equation of L1 .
L1 meets the curve again at the point Q .
b) Determine the coordinates of Q .
The tangent to C at Q is the straight line L2 .
L2 meets the y axis at the point R .
c) Show that the area of the triangle PQR is one square unit.
C1B , y = x + 3 , Q (1, 4 )
Created by T. Madas
Created by T. Madas
Question 199
(****)
The figure below shows the design of a hazard warning logo which consists of three
identical sectors of radius r cm, joined together at the centre.
Each sector subtends an angle θ radians at the centre and the sectors are equally
spaced so that the logo has rotational symmetry of order 3 .
θ
r
The area of the logo is 75 cm 2 .
a) Show that the perimeter P cm of the logo is given by
P = 6r +
150
.
r
b) Determine by differentiation the value of r for which P is stationary.
c) Show that the value of r found in part (b) gives the minimum value for P .
d) Find the minimum perimeter of the feeder.
r = 5 , Pmin = 60
Created by T. Madas
