For f(x)=3x4−12x3+5 find the following.(A) f′(x)(B) The slope of the graph of f at x=2(C)...

Talamancoeb

Talamancoeb

Answered

2021-12-31

For f(x)=3x412x3+5 find the following.
(A) f(x)
(B) The slope of the graph of f at x=2
(C) The equation of the tangent line at x=2
(D) The value(s) of x where the tangent line is horizontal
(A) f'(x) = ____
(B) At x=2, the slope of the graph of f is ___
(C) At x=2, the equation of the tangent line is y = ___
(D) The tangent line is horizontal at x =_____

(Use a comma to separate answers as needed.)

Answer & Explanation

Andrew Reyes

Andrew Reyes

Expert

2022-01-01Added 24 answers

Step 1
Given:
f(x)=3x412x3+5
Step 2
A) To find f(x)
Differentiate the function with respect to x
ddxf(x)=ddx(3x412x3+5)
f(x)=12z336x2
Elaine Verrett

Elaine Verrett

Expert

2022-01-02Added 41 answers

Step 3
B) The slope of the f(x) at x=2
Substitute x=2 in f(x)
f(x)x=2=12(2)336(2)2
=12(8)36(4)
=96144
=48
Step 4
Therefore, the slope of f at x=2 is -48.
karton

karton

Expert

2022-01-04Added 439 answers

Step 5
NSK
C) The equation of the tangent line at x = 2
First find the y coordinates by substituting x=2 in the given function.
f(2)=3(2)412(2)3+5
y=48-96+5
y=-43
Step 6
Therefore, the equation of the tangent line passing through (2,-43) with slope -48, is
y-(-43)=-48(x-(-2))
y+43=-48x+96
y=-48x+96-43
y=-48x+53
Step 7
D)
The value(s) of x where the tangent line is horizontal
Set f(x) equal to 0 and solve for x to find the x values at which the tangent line is horizontal to given function.
12x336x2=0
12x2(x3)=0
12x2=0,(x3)=0
x=0,(x3)=0
x=0,x=3
Step 8
Therefore, the tangent line is horizontal at x=0,3

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