91Ó°ÊÓ

The Intermediate Value Theorem (IVT) is a fundamental concept in calculus. It states that for any continuous function \( f \) that spans between two values \( f(a) \) and \( f(b) \) on a closed interval \([a, b]\), the function will take on every value between \( f(a) \) and \( f(b) \) at some point within that interval.
This theorem is crucial in proving the existence of a root or a specific value within an interval. It is applicable when you need to find out if a function crosses a particular value within a specific range.
In our exercise, IVT helps confirm that if \( f'(a) \cdot f'(b) = -1 \), then there's a point \( c \) where the derivative \( f'(c) = 0 \). This occurs because the IVT ensures that \( f'(x) \), a continuous function, must pass through all intermediate values between \( f'(a) \) and \( f'(b) \) as it moves from one end of the interval to the other. Thus, if the sign changes, zero is one of those values.

Understanding tangent and normal lines is essential in differential calculus. A tangent line to a curve at a given point has the same slope as the curve at that point. If \( y = f(x) \), the slope of the tangent line at any point \( x = a \) is the derivative \( f'(a) \).
The equation of a tangent line at \( x = a \) can is given by:

• \( y - f(a) = f'(a)(x - a) \)

This line touches the curve at exactly one point, without cutting across it.
On the other hand, a normal line is perpendicular to the tangent line at a given point on the curve. This means the slope of the normal line at point \( x = b \) is the negative reciprocal of the slope of the tangent line, which is \(-\frac{1}{f'(b)}\).
In our exercise, the tangent line at \( x = a \) is also the normal line at \( x = b \). Hence we establish the unique condition :

• \( f'(a) = -\frac{1}{f'(b)} \), leading to \( f'(a) \cdot f'(b) = -1 \)

Continuous and differentiable functions have smooth graphs without breaks or sharp turns.
A function \( f \) is continuous at a point if the limit of \( f \) as it approaches that point from both directions is the same as the function's value at that point.
Differentiability relates to continuity but adds the requirement that the function's rate of change (slope) is consistent and predictable. This is expressed using derivatives. If a function is differentiable at a point, it is also continuous at that point. However, a function can be continuous but not differentiable.
In the exercise, \( f \) being continuous and differentiable ensures the Intermediate Value Theorem's applicability. It confirms that \( f'(x) \), the derivative, is continuous. Thus, any changes in \( f'(x) \) across interval \((a, b)\) are smooth, allowing us to identify at least one point \( c \) within \((a, b)\) where \( f'(c) = 0 \). This continuity ensures \( f \) does not "jump over" potential roots or slopes.