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Consider the cubic function f(x) = ax^3 + bx^2 + cx + d. Determine the values of a, b, c, and d so that all of the following conditions are met.
a. f '(-1) = 1, f '(0) = -2
b. ๐โ๐๐๐ ๐๐ ๐ ๐๐๐๐๐ก ๐๐ ๐๐๐๐๐๐๐ก๐๐๐ ๐๐ก (1,0)
c. ๐โ๐ ๐ฆ ๐๐๐ก๐๐๐๐๐๐ก ๐๐ ๐กโ๐ ๐๐ข๐๐๐ก๐๐๐ ๐ฆ = ๐(๐ฅ) ๐๐ (0, โ3)
Find the constants ๐, ๐, ๐, d
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The condition (c) gives us d = f(0) = -3.
So, the value of d is joust found.
The condition (a) f'(-1) = 1 leads to this equation
f'(-1) = 3a*(-1)^2 + 2b(-1) + c = 1, or
3a - 2b + c = 1. (1)
The condition (a) f'(0) = -2 leads to this equation
f'(0) = 3a*0^2 + 2b*0 + c = -2, or
c = -2. (2)
So, the value of c is just found: c = -2.
The condition (b) means that the second derivative of f(x) has zero value at the point x= 1
f''(1) = 0, or
6a + 2b = 0. (3)
By the way, the condition (b) also means that the point (1,0) lies on the curve y = f(x), or f(1) = 0.
It is a "hidden", an ADDITIONAL and an EXCESSIVE condition, which makes the solution impossible and non-existing, as you will see it later.
So, we just have two equation to determine two remaining unknown coefficients "a" and "b"
3a - 2b = 3 (4) (obtained from (1) and (2))
6a + 2b = 0 (5) (it is equation (3))
To solve this system, add equations (4) and (5). You will get then
9a = 3,
which implies a = .
Then from (4), 2b = - = -2; hence, b = -1.
So, we just found all coefficients a = ; b = -1; c = -2 and d = -3.
y = f(x) = .
CHECK
(a) f'(-1) = 3*(1/3)*(-1)^2 - 2*(-1) - 2 = 1 + 2 - 2 = 1. ! Correct !
(a) f'(0) = -2. ! Correct !
(b) f''(1) = 3*2*(1/3) - 2*1 = 2 - 2 = 0 ! Correct !
Now I should check the "hidden" condition that the point (1,0) lies on the curve.
f(1) = 3*(1/3) - 2*1 - 2*1 - 3 = 1 - 2 - 2 - 3 = -6 =/= 0 ! Incorrect !
The conclusion. As worded and presented, the given problem is self-contradictory and has no solution.
More explanations and post-solution note
Had the problem said "there is a point of inflection at x = 1", the problem would have a solution as shown above.
But in the form "there is a point of inflection at (1,0)", it brings a "hidden" condition that the point (1,0) lies
on the curve. With this hidden condition, we have 4+1 = 5 equations for four unknowns.
These 5 equations are INCONSISTENT, which makes the solution in this form IMPOSSIBLE and NON-EXISTING.
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