Question

1
Determine the value of k if g(x)=4x+k is a tangent to f(x)=-x²+8x+20​

Answer 1

Answer:

k=24

Step-by-step explanation:

The tangent of the function f at x=a, can be found by differentiating f w.r.t. x and then replacing x with a.

f=-x^2+8x+20

Differentiating both sides:

f'=(-x^2+8x+20)'

By sum rule:

f'=(-x^2)'+(8x)'+(20)'

By constant multiple rule:

f'=-(x^2)'+8(x)'+(20)'

By constant rule:

f'=-(x^2)+8(x)'+0

By power rule:

f'=-2x+8

f' at x=a is -2a+8

This is the slope of any tangent line to the curve f.

The slope of g is 4 if you compare it to slope intercept form y=mx+b.

So we gave -2a+8=4.

Subtracr 8 on both sides: -2a=-4

Divide both sides by -2: a=2

The tangent line to the curve at x=2 is y=4x+k.

To tind y we must first know the y-coordinate of the point of tangency.

If x=2, then

f(2)=-(2)^2+8(2)+20=-4+16+20=12+20=32

So the point is (2,32).

g(x)=4x+k and we know g(2)=32.

This gives us:

32=4(2)+k

32=8+k

k=32-8

k=24

Answer 2

From the tangent given, the value of k will be 24.

How to solve the tangent

From the information given, g(x) = 4x + k and this is a tangent to f(x)= -x² +8x + 20.

f(x)= -x² +8x + 20

= f'(x)  = -2x  + 8

The slope of tangent = -2x + 8  at (x , y)

g(x) = 4x  + k  

Hence, slope = 4

-2x + 8 = 4

Collect like terms

-2x = 4 - 8

-2x = - 4

Divide both side by -2

-2x/-2 = -4/-2

x = 2

f(x)= -x² +8x +20

where, x = 2

y = -2² + 8(2) + 20  

y =  32

Hence (2 , 32) lies on g(x)= 4x+k

32 = 4(2) + k

32 = 8 + k

k = 32 - 8

k = 24

In conclusion, the value of k is 24.

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