Question

If \(F\) is a differentiable function such that \(F(3) = 6\) and \(F(9) = 2\), then there must exist at least one number 'a' between 3 and 9, such that:

• A. \(F'(a) = \dfrac{3}{2}\)
• B. \(F(a) = -\dfrac{3}{2}\)
• C. \(F'(a) = -\dfrac{3}{2}\)
• D. \(F'(a) = -\dfrac{2}{3}\)

Hint:

Use the Lagrange Mean Value Theorem: the derivative at some point between 3 and 9 must equal the average rate of change (F(9)-F(3))/(9-3).

Answer 1

The Correct Option is D

Let's think about this using the geometric picture behind the Mean Value Theorem, instead of just quoting the formula.

Picture the graph of $F$ between $x=3$ and $x=9$. We know the graph passes through the point $(3, 6)$ and the point $(9, 2)$. Draw the straight line joining these two points; this is called the chord. Its slope is the average rate at which $F$ changes over this stretch.

1. Slope of the chord: rise over run gives $\frac{2-6}{9-3} = \frac{-4}{6} = -\frac{2}{3}$. This single number describes how the two endpoints compare, regardless of what $F$ does in between.
2. What LMVT adds: because $F$ is differentiable everywhere on $(3,9)$, so its graph has no corners or breaks, somewhere along that stretch the curve must have a tangent line exactly parallel to this chord, meaning the same slope as the chord.
3. Why this must happen: if the curve's slope were always steeper than the chord's slope, $F$ would change faster than the chord the whole way and overshoot the endpoint value; if it were always shallower, it would undershoot. So the curve's slope has to match the chord's slope at least once strictly inside the interval.

Since the tangent slope at that point $a$ is $F'(a)$, and we found the chord's slope is $-\frac{2}{3}$, we get $F'(a) = -\frac{2}{3}$.

Let's summarize:

• The chord joining $(3,6)$ and $(9,2)$ has slope $-\frac{2}{3}$.
• LMVT guarantees a point where the tangent to $F$ is parallel to this chord.
• Options naming a function value instead of a derivative, or giving a different number, do not match this slope.

So the correct statement is $F'(a) = -\frac{2}{3}$, which is option (D).