which of the following is an equation of the line tangent to the graph of $g(x)=2x^{3}+5x^{2}$ at the point…

which of the following is an equation of the line tangent to the graph of $g(x)=2x^{3}+5x^{2}$ at the point where $x=-2$?\nchoose 1 answer:\na $y = 4x + 12$\nb $y = 4x - 18$\nc $y = 4x - 12$\nd $y = -2x + 12$

which of the following is an equation of the line tangent to the graph of $g(x)=2x^{3}+5x^{2}$ at the point where $x=-2$?\nchoose 1 answer:\na $y = 4x + 12$\nb $y = 4x - 18$\nc $y = 4x - 12$\nd $y = -2x + 12$

Answer

Explanation:

Step1: Find the derivative of (g(x))

Using the power rule ((x^n)^\prime = nx^{n - 1}), for (g(x)=2x^{3}+5x^{2}), (g^\prime(x)=6x^{2}+10x).

Step2: Find the slope of the tangent line at (x = - 2)

Substitute (x=-2) into (g^\prime(x)): (g^\prime(-2)=6\times(-2)^{2}+10\times(-2)=6\times4 - 20=24 - 20 = 4).

Step3: Find the (y) - coordinate of the point of tangency

Substitute (x = - 2) into (g(x)): (g(-2)=2\times(-2)^{3}+5\times(-2)^{2}=2\times(-8)+5\times4=-16 + 20 = 4). So the point of tangency is ((-2,4)).

Step4: Use the point - slope form (y - y_1=m(x - x_1))

Here (m = 4), (x_1=-2), (y_1 = 4). (y - 4=4(x + 2)), expand it: (y-4=4x + 8), then (y=4x+12).

Answer:

A. (y = 4x + 12)