The line tangent to the graph of g(x) = x^{3}-4x+1 at the point (2, 1) is given by the formula

L(x) = 8(x - 2) + 1.

Which of the following statements are true? Select all that apply.
A. The functions are approximately equal on the interval 1.9 smaller than or equal to x larger than or equal to 2.1.
B. The functions have the same y-intercept.
C. The slope of the line equals g'(0).
D. The functions are approximately equal on the interval 0.5 smaller than or equal to x larger than or equal 1.0.
E. The graphs intersect at the point (2,1).
F. The slope of the line equals g'(2).
G. None of the above are true.

Using function concepts, it is found that the correct options are:

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A similar problem is given at link

well, about A and D, I just plugged the values on the slope formula of

for A the values are 8.01 and 8.0, so indeed those "slopes" are close.

for D the values are -2.25 and 8.0, so no dice on that one.

for B, let's check the y-intercept for g(x), by setting x = 0, we end up g(0) = 0³-4(0)+1, which gives us g(0) = 1.

checking L(x) y-intercept, well, L(x) is in slope-intercept form, thus the +1 sticking out on the far right is the y-intercept, so, dice.

for C, well, the slope if L(x) is 8, since it's in slope-intercept form, the derivative of g(x) is g'(x) = 3x² - 4, and thus g'(0) = -4, so no dice.

for E, do they intercept at (2,1)?  well, come on now, L(x) is a tangent line to g(x), so that's a must for a tangent.

for F, we know the slope of the line L(x) is 8, is g'(2) = 8?  let's check

recall that g'(x) = 3x² - 4, so g'(2) = 3(2)² - 4, meaning g'(2) = 8, so, dice.