Question

Help please. Hard calculus question :(

Answer 1

@dumbcow @satellite73 help please

Answer 2

I have gotten (a) to be
\[a(2x - \alpha - \beta)\]

Answer 3

correct

Answer 4

now i have no idea what do to for part (b) and (c)

Answer 5

ok
x-intercepts are alpha and beta
plug those in for "x" into derivative to get tangents

-->dy/dx = a(alpha -beta)
--> dy/dx = a(beta - alpha) = -a(alpha -beta)

tangents are indeed opposite signs

Answer 6

ahh ok. So we make the 2 tangents equal each other to find point M?

Answer 7

yes but we need equation of tangent lines .... above is just the slope of tangent line
but using given point (the x-intercept) you can get each equation from point-slope form
\[ y-0 = a(\alpha -\beta)(x-\alpha)\]
\[ y-0 = -a(\alpha -\beta)(x-\beta)\]
set equations equal , solve for x to find point M

Answer 8

as problem states you should find
\[x = \frac{\alpha +\beta}{2}\]

Answer 9

So when the tangent is horizontal, that means the gradient = 0 right?

Answer 10

once we get\[x= \frac{ \alpha + \beta }{ 2}\] we need to sub it into y= a(x-alpha)(x-beta) to find the y coordinate?

Answer 11

ok yes set equal to 0.... you should get same x-value for both V and M

then plug back into original "y" to get V
plug into tangent line equation to get M

Answer 12

we need to sub it into y= a(x-alpha)(x-beta) to find the y coordinate?

Answer 13

correct, that will give y-coord of V

Answer 14

Oh, why isnt it y-coordinate of M?

Answer 15

V is point on curve where tangent is horizontal ... the vertex

M is point where 2 tangent lines intersect

Answer 16

So point M lies on the x intercepts?