Question

Determine the value of life g(x)=4x+k is a tangent to f(x)=-x^2+8x+20​

Answer 1

k=16

Explanation:

So the tangent line is

`4x + k`

and it tangent to function

`{x}^(2) + 8x + 20`

Since the slope of the tangent line is 4, this means the derivative of f(x) is 4 but first let find the derivative of

`{x}^(2) + 8x + 20`

Use the Sum Rule,

`(d)/(dx) {x}^(2) + (d)/(dx) 8x + (d)/(dx) 20`

Use the Power Rule and we get

`2x + 8`

Set this equal to 4

`2x + 8 = 4`

`2x = - 4`

`x = - 2`

So at x=-2, the slope of the tangent line is 4.

Plug -2 in the orginal function, and we get

`{ - 2}^(2) + 8( - 2) + 20 = 8`

So the point must pass through -2,8 with a slope of 4.

`y - 8 = 4(x + 2)`

`y - 8 = 4x + 8`

`y = 4x + 16`

So the value of k is 16.