Consider the cubic function f(x) = ax^3 + bx^2 + cx + d.  Determine the values of the constants a, b, c and dso that f(x) has a point of inflection at the origin and a local maximum at the point...

Consider the cubic function f(x) = ax^3 + bx^2 + cx + d.  Determine the values of the constants a, b, c and d

so that f(x) has a point of inflection at the origin and a local maximum at the point (2, 4).

Asked on by bahadur

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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`f(x)= ax^3 + bx^2 + cx + d`

f(0)= 0

==> d = 0

f(2)= 4

==> 8a + 4b + 2c = 4

==> 4a + 2b + c = 2 .............(1)

`f'(x)= 3ax^2 + 2bx + c` 

f'(2)= 0

==> 12a + 4b + c = 0...........(2)

8a + 2b = -2

4a + b = -1 ..................(3)

f''(x)= 6ax + 2b

f''(0)= 0

==> b = 0

4a = -1

==> `a = -1/4`

`` ==> 4a + c = 2

==> -1 + c = 2

==> c = 3

`==gt f(x)= (-1/4)x^3 + 3x` `==gt a = -1/4 ==gt b= 0 ==gt c = 3 ==gt d = 0`

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