Need math help..? Mean Value Theorem?

Question

Can someone help me with a few true and false math problems?


1.Let f be continuous on the closed interval [ab] and differentiable on the open interval (ab), and let A=(af(a)) and B=(bf(b)). Then there is at least one point P on the graph of f where the tangent line is parallel to the secant line AB.

2.The Mean Value Theorem is a special case of Rolle's Theorem.

3.Suppose that f and g are continuous on [ab] and differentiable on (ab). If f(x)=g(x) for all x in (ab), then f and g differ by a constant.

4.If f is continuous on the closed interval [ab] and differentiable on the open interval (ab) then there is a number c in (ab) such that f(b)−f(a)=f(c)(b−a).

5.If f is differentiable on the set of real numbers and has two roots, then f has at least one root. 

I have down:

1. True
2. False
3. True
4. True
5. False

Can someone tell me what I'm doing wrong?

mecdub · Answer

3, 4, and 5 are wrong.

3. If f(x) = g(x) on (a, b) then they don't differ at all on the interval. If the derivatives f '(x) = g'(x), then f and g differ by a constant.

4. The MVT relates the slope of the secant line through a and b to the slope of the tangent line at c. The one in this question relates the secant line's slope to the value of f at c. The correct formula is f(b) - f(a) = f '(c)(b - a).

5. If f has two roots, then clearly it has at least one root. Looking at this one makes me wonder if your prime symbols just aren't showing up on my screen. If that's the case, then 3 and 4 may be correct and only this one is wrong. Either way, if that was supposed to say "the f ' has at least one root" Rolle's Theorem proves that to be true.

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Need math help..? Mean Value Theorem?

1 Answer