homework4: problem 12\n(1 point)\nlet $f(x)=3x(5 - 4x)^2$.\nfind the equation of line tangent to the graph…

homework4: problem 12\n(1 point)\nlet $f(x)=3x(5 - 4x)^2$.\nfind the equation of line tangent to the graph of $f$ at $x=-1$\ntangent line: $y = $\npreview my answers submit answers\nyou have attempted this problem 0 times.\nyou have unlimited attempts remaining.\nemail instructor

homework4: problem 12\n(1 point)\nlet $f(x)=3x(5 - 4x)^2$.\nfind the equation of line tangent to the graph of $f$ at $x=-1$\ntangent line: $y = $\npreview my answers submit answers\nyou have attempted this problem 0 times.\nyou have unlimited attempts remaining.\nemail instructor

Answer

Explanation:

Step1: Expand the function

First, expand (f(x)=3x(5 - 4x)^{2}=3x(25 - 40x+16x^{2}) = 75x-120x^{2}+48x^{3}).

Step2: Find the derivative

Using the power - rule ((x^n)^\prime=nx^{n - 1}), (f^\prime(x)=75-240x + 144x^{2}).

Step3: Find the slope of the tangent line

Evaluate (f^\prime(x)) at (x=-1). (f^\prime(-1)=75+240 + 144=459).

Step4: Find the y - coordinate of the point

Evaluate (f(x)) at (x = - 1). (f(-1)=3\times(-1)\times(5 + 4)^{2}=-3\times81=-243).

Step5: Use the point - slope form

The point - slope form of a line is (y - y_1=m(x - x_1)), where ((x_1,y_1)=(-1,-243)) and (m = 459). So (y+243=459(x + 1)), which simplifies to (y=459x+459 - 243=459x+216).

Answer:

(y = 459x+216)