For what values of
x
does the function
f
(
x
)
=
12
x
−
x
3
have a horizontal tangent line?

Select all that apply.

Question

For what values of 
x
  does the function 
f
(
x
)
=
12
x
−
x
3
  have a horizontal tangent line?

Select all that apply.

BlackTom AI · Accepted Answer

To locate horizontal tangent lines, we first compute the derivative of f(x) = 12x − x^3 and set it equal to zero where the slope is zero. f'(x) = 12 − 3x^2. Setting f'(x) = 0 gives 12 − 3x^2 = 0, which simplifies to 3x^2 = 12 and x^2 = 4, hence x = ±2. This means horizontal tangents occur at x = -2 and x = 2. Now evaluate each provided option to see whether it matches these values and contributes to a horizontal tangent.

- Option -1: f'(-1) = 12 − 3(−1)^2 = 12 − 3 = 9 ≠ 0, so not a horizontal tangent point. 
- Option 3: f'(3) = 12 − 3(3)^2 = 12 − 27 = −15 ≠ 0, so not a horizontal tangent. 
- Option -4: f'(-4) = 12 − 3(16) = 12 − 48 = −36 ≠ 0, not a horizontal tangent. 
- Option -12: f'(-12) = 12 − 3(144) = 12 − 432 = −420 ≠ 0, again not zero. 
- Option 2: f'(2) = 12 − 3(4) = 12 − 12 = 0, this is a horizontal tangent value. 
- Option 1: f'(1) = 12 − 3(1) = 9 ≠ 0, not a horizontal tangent. 
- Option 0: f'(0) = 12 − 3(0) = 12 ≠ 0, not a horizontal tangent. 
- Option 4: f'(4) = 12 − 3(16) = 12 − 48 = −36 ≠ 0, not a horizontal tangent. 
- Option -3: f'(-3) = 12 − 3(9) = 12 − 27 = −15 ≠ 0, not a horizontal tangent. 
- Option -2: f'(-2) = 12 − 3(4) = 0, this is a horizontal tangent value. 
- Option 12: f'(12) = 12 − 3(144) = 12 − 432 = −420 ≠ 0, not a horizontal tangent.

In summary, only the x-values that produce a zero derivative are x = −2 and x = 2, so among the given options the correct horizontal-tangent values are -2 and 2.

For what values of x does the function f ( x ) = 12 x − x 3 have a horizontal tangent line? Select all that apply. 多项选择题

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