Question

Guys please help My is assignment due tomorrow please guyyyysssss. Determine the value of k if g(x) = 4x+k is a tangent to f(x) = - x² + 8x +20​

Answer 1

so, since we know that the derivative is simply the equation we use to get the slope at any point on the curve, thus df/dx will be that.

now, we know that g(x) is already in slope-intercept form, so it has a slope of "4", hmmm what's "x" at that instance?

`f(x)=-x^2+8x+20\implies \cfrac{df}{dx}=-2x+8 \\\\[-0.35em] ~\dotfill\\\\ g(x)=\stackrel{\stackrel{m}{\downarrow }}{4}x+k\qquad \impliedby \qquad \begin{array}c \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array} \\\\\\ 4~~ = ~~-2x+8\implies 2=x`

well, now we know that x = 2, hmmm what's "y" at that instance? well, our "y" will be simply f(2) = g(x), because g(x) is touching f(x) at that point on the curve.

`f(2)=-(2)^2+8(2)+20\implies f(2)=32 \\\\[-0.35em] ~\dotfill\\\\ g(x)=4x+k\implies 32=4(2)+k\implies 32=8+k\implies \boxed{24=k}`

Check the picture below.