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

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

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

Answer

Explanation:

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

Using the power rule ((x^n)^\prime = nx^{n - 1}), we have (h^\prime(x)=5x^{4}+6x + 2).

Step2: Evaluate (h(-1)) and (h^\prime(-1))

  • For (h(-1)): (h(-1)=(-1)^{5}+3(-1)^{2}+2(-1)=-1 + 3-2=0).
  • For (h^\prime(-1)): (h^\prime(-1)=5(-1)^{4}+6(-1)+2=5 - 6 + 2=1).

Step3: Use the point - slope form (y - y_1=m(x - x_1))

Here (x_1=-1,y_1 = 0,m = 1). Substituting into (y - y_1=m(x - x_1)), we get (y-0=1\times(x + 1)), which simplifies to (y=x + 1).

Answer:

D. (y = x + 1)