Answer is: option3

1.5

Solution:

We are given the function:

\[ f(x) = 3x^2 - 4x + 2 \] Find \( f(1) \) \[ f(1) = 3(1)^2 - 4(1) + 2 = 3 - 4 + 2 = 1 \] Find \( f'(x) \)

Differentiate \( f(x) \):

\[ f'(x) = \frac{d}{dx} (3x^2 - 4x + 2) = 6x - 4 \]

Now, evaluate \( f'(1) \):

\[ f'(1) = 6(1) - 4 = 6 - 4 = 2 \] Equation of the Tangent Line at \( x = 1 \)

The tangent line follows the point-slope formula:

\[ y - f(1) = f'(1)(x - 1) \]

Substituting values:

\[ y - 1 = 2(x - 1) \] \[ y = 2x - 2 + 1 \] \[ y = 2x - 1 \]

So, the tangent line equation is:

\[ L(x) = 2x - 1 \] Finding the Smallest \( x \) for \( |f(x) - L(x)| > 0.5 \)

From the computed table, we observe that the error \( |f(x) - L(x)| \) exceeds 0.5 for the first time at \( x = 1.5 \).

x	f(x)	L(x)	|f(x) - L(x)|
1.3	1.87	1.6	0.27
1.4	2.28	1.8	0.48
1.5	2.75	2.0	0.75
1.6	3.28	2.2	1.08
1.7	3.87	2.4	1.47

Final Answer

Thus, the smallest value of \( x \) for which the error is more than 0.5 is: 1.5