Question

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

Answer 1

Answer:

k=16

Step-by-step 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,

$\frac{d}{dx} {x}^{2} + \frac{d}{dx} 8x + \frac{d}{dx} 20$

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.