For the function f(x)=x^3-2x^2+2x, what is (f^-1)'(4)?
A. 1/4
B. 1/6
C. 1/34
D. 6
E. 34
f(x)=x^3-2x^2+2x
y = x^3-2x^2+2x
the inverse is:
x = y^3 - 2y^2 + 2y
take the derivative , that's what f^-1 ' (x) means
1 = 3y^2 dy/dx - 4y dy/dx + 2dy/dx
so when y = 4
1 = 3(16) dy/dx - 16 dy/dx + 2 dy/dx
1 = 34 dy/dx
dy/dx = 1/34
@mathhelper
the way you did makes sense but here's how i did it and ended up getting a different ans:
since (f^-1)'(4), you know f(x)=4, so:
x^3-2x+2x=4
x=2
since (f^-1)'(x)=1/f'(f^-1(x)):
we can find the derivative of f(x):
f'(x)=3x^2+4x+2
solve for x=2:
f'(x)=3x^2+4x+2
f'(x)=6
since (f^-1)'(x)=1/f'(f^-1(x)):
(f^-1)'(x)=1/6
both of these methods seem right to me, but im not sure which answer is correct one?
Recall that if g(x) is the inverse of f(x) then
if f(a) = b then g'(b) = 1/f'(a)
f(2) = 4
so g'(4) = 1/f'(2) = 1/6
To find the derivative of the inverse function, we first need to find the inverse function of f(x).
To find the inverse function, we switch the roles of x and y in the original function and solve for y:
x = y^3 - 2y^2 + 2y
Next, we solve this equation for y.
Rearranging the terms, we have:
y^3 - 2y^2 + 2y - x = 0
Now, we need to solve this cubic equation for y in terms of x. However, finding the inverse function directly can be a complicated process.
Alternatively, we can use the fact that the derivative of the inverse function is the reciprocal of the derivative of the original function evaluated at the corresponding point.
The derivative of f(x) = x^3 - 2x^2 + 2x is given by:
f'(x) = 3x^2 - 4x + 2
To find (f^-1)'(4), we need to evaluate f'(x) at the corresponding point of x = 4 in the inverse function.
So, we need to find the value of x such that f(x) = 4.
Setting f(x) = 4, we have:
4 = x^3 - 2x^2 + 2x
Rearranging the terms, we have:
x^3 - 2x^2 + 2x - 4 = 0
Solving this cubic equation will give us the corresponding value of x. However, solving cubic equations can be complicated, and the solution may not be easily obtainable.
Given the options provided, we can calculate the derivative of f(x), f'(x), for each option, and see which one matches the value we obtain.
Let's calculate f'(x) for each option and see which one matches our calculated value:
A. 1/4:
f'(1/4) = 3(1/4)^2 - 4(1/4) + 2 = 1/4 - 1 + 2 = 7/4
B. 1/6:
f'(1/6) = 3(1/6)^2 - 4(1/6) + 2 = 1/12 - 2/3 + 2 = 11/12
C. 1/34:
f'(1/34) = 3(1/34)^2 - 4(1/34) + 2 = 1/3400 - 4/34 + 2 = -1103/1700
D. 6:
f'(6) = 3(6)^2 - 4(6) + 2 = 108 - 24 + 2 = 86
E. 34:
f'(34) = 3(34)^2 - 4(34) + 2 = 3468 - 136 - 4 = 3428
From the calculated values, the closest match to the value we obtained for (f^-1)'(4) is option A. 1/4.
Therefore, the answer is A. 1/4.