Question

Let f be the function defined by f(x) = 4x^3 - 5x + 3. Which of the following is an equation of the line tangent to the graph of f at the point where x = -1 ?

Answer 1

The equation of the line tangent to the graph of f defined by g(x) = 4x^3 - 5x + 3 at the point where x = -1 is y = 7x + 11

Given the function of the graph defined as gx) = 4x^3 - 5x + 3.

If x = -1

f(-1) = 4(-1)^3 - 5(-1) + 3.

f(-1) = 4(-1) + 5 + 3

f(-1) = -4 + 8

f(-1) = 4

Hence the coordinate point on the line will be (-1, 4)

Get the slope of the line at x = 1

$m=\frac{dy}{dx} = 12x^2-5\\m=12(-1)^2-5\\m=12-5\\m=7$

Get the required equation. The equation of the line in point-slope form is expressed as $y-y_0=m(x-x_0)$

$y-4=7(x+1)\\y-4=7x+7\\y=7x+7+4\\y=7x+11$

Hence the equation of the line tangent to the graph of f at the point where x = -1 is y = 7x + 11

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