Curve Sketching Problem (1 Viewer)

[the-derivative]

Hey guys,

I was going through my school's past papers and I found this question. It makes sense to me, but I don't know how to explain it properly to get the 5-marks. I can get the sketch, but I was wondering if you guys could help me explain it.

Question (5 marks):

Rolle's Theorem states "If y = f(x) is a smooth continuous function and if f(a) = f(b) for a[FONT=&quot] ϵ[/FONT] R and b[FONT=&quot] ϵ[/FONT] R, then there exists a numner, m, between a and b such that f'(m) = 0.

Illustrate this theorem with a sketch and brief explanation.

 

[Timothy.Siu]

Let a< b If f(a)=f(b) and f(x) is a continuous function then every point of the function is differentiable. The function can either increase, decrease or stay the same after reaching point A. If the function increases after reaching point a, then it would have to decrease again before reaching point b, which would require a turning point at a point m, and f'(m)=0 since it is a continuous smooth function. Similarly, for decreasing after point A.
However if it stays the same after point A, f(m)=f(a)=f(b) the function is a straight line, y=P and y'=0 thus, f'(m)=0 adhering to the theorem.

haha thats what i would say

 

[shinn]

Rolle's theorem - Wikipedia, the free encyclopedia

 

[the-derivative]

Cool, thanks guys.