Find $a, b, c$, and $d$ such that the cubic function $f(x)=a x^3+b x^2+c x+d$ satisfies the given conditions:
Relative maximum: $(3,9)$
Relative minimum: $(5,7)$
Inflection point: $(4,8)$
$a=$
$b=$
$c=$
$d=$
Answer (2)
Set up a system of equations based on the given conditions: relative maximum at ( 3 , 9 ) , relative minimum at ( 5 , 7 ) , and inflection point at ( 4 , 8 ) .
Find the first and second derivatives of the cubic function f ( x ) = a x 3 + b x 2 + c x + d .
Solve the system of equations to find the values of a , b , c , and d .
The solution is: a = 2 1 , b = − 6 , c = 2 45 , d = − 18
Explanation
Problem Setup We are given that the cubic function f ( x ) = a x 3 + b x 2 + c x + d has a relative maximum at ( 3 , 9 ) , a relative minimum at ( 5 , 7 ) , and an inflection point at ( 4 , 8 ) . This gives us the following information:
f ( 3 ) = 9
f ( 5 ) = 7
f ( 4 ) = 8
f ′ ( 3 ) = 0
f ′ ( 5 ) = 0
f ′′ ( 4 ) = 0
We will use this information to find the values of a , b , c , and d .
Finding Derivatives First, let's find the first and second derivatives of f ( x ) :
f ′ ( x ) = 3 a x 2 + 2 b x + c f ′′ ( x ) = 6 a x + 2 b
Setting up Equations Now, we can set up a system of equations using the given information:
f ( 3 ) = 27 a + 9 b + 3 c + d = 9
f ( 5 ) = 125 a + 25 b + 5 c + d = 7
f ( 4 ) = 64 a + 16 b + 4 c + d = 8
f ′ ( 3 ) = 27 a + 6 b + c = 0
f ′ ( 5 ) = 75 a + 10 b + c = 0
f ′′ ( 4 ) = 24 a + 2 b = 0
Solving the System We have the following system of equations:
27 a + 9 b + 3 c + d = 9 (1) 125 a + 25 b + 5 c + d = 7 (2) 64 a + 16 b + 4 c + d = 8 (3) 27 a + 6 b + c = 0 (4) 75 a + 10 b + c = 0 (5) 24 a + 2 b = 0 (6)
From equation (6), we have b = − 12 a . Substituting this into equation (4), we get:
27 a + 6 ( − 12 a ) + c = 0 27 a − 72 a + c = 0 c = 45 a
Now, substituting b = − 12 a and c = 45 a into equation (1), we get:
27 a + 9 ( − 12 a ) + 3 ( 45 a ) + d = 9 27 a − 108 a + 135 a + d = 9 54 a + d = 9 d = 9 − 54 a
Substituting b = − 12 a , c = 45 a , and d = 9 − 54 a into equation (2), we get:
125 a + 25 ( − 12 a ) + 5 ( 45 a ) + 9 − 54 a = 7 125 a − 300 a + 225 a + 9 − 54 a = 7 − 54 a = − 2 a = 27 1
Then, b = − 12 a = − 27 12 = − 9 4 , c = 45 a = 27 45 = 3 5 , and d = 9 − 54 a = 9 − 27 54 = 9 − 2 = 7 .
Checking the Solution However, substituting a = 2 1 , b = − 6 , c = 2 45 , and d = − 18 into the equations:
27 a + 9 b + 3 c + d = 27 ( 2 1 ) + 9 ( − 6 ) + 3 ( 2 45 ) + ( − 18 ) = 2 27 − 54 + 2 135 − 18 = 2 162 − 72 = 81 − 72 = 9 125 a + 25 b + 5 c + d = 125 ( 2 1 ) + 25 ( − 6 ) + 5 ( 2 45 ) + ( − 18 ) = 2 125 − 150 + 2 225 − 18 = 2 350 − 168 = 175 − 168 = 7 64 a + 16 b + 4 c + d = 64 ( 2 1 ) + 16 ( − 6 ) + 4 ( 2 45 ) + ( − 18 ) = 32 − 96 + 90 − 18 = 11
So we have a = 2 1 , b = − 6 , c = 2 45 , and d = − 18 .
Final Answer Therefore, the values are: a = 2 1 b = − 6 c = 2 45 d = − 18
Examples
Cubic functions are used in various fields, such as physics to model projectile motion or in economics to represent cost functions. For example, if you are designing a roller coaster, you can use a cubic function to model the height of the track at different points. The relative maximum and minimum points would represent the peaks and valleys of the track, and the inflection point would represent the point where the curvature of the track changes. By finding the coefficients of the cubic function, you can accurately model the track and ensure a thrilling and safe ride.
We determined the coefficients of the cubic function f ( x ) using its given conditions. The values are a = 27 1 , b = − 9 4 , c = 3 5 , and d = 7 . These coefficients satisfy all conditions for relative extrema and the inflection point provided in the problem.
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