Problem 48

Question

Use the Mean Value Theorem to show that the graph of a concave up function $f$ is always above its tangent line; that is, show that $$ f(x)>f(c)+f^{\prime}(c)(x-c), \quad x \neq c $$

Step-by-Step Solution

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Answer

The graph of a concave up function is above its tangent line, as \$ f(x) > f(c) + f'(c)(x-c) \$.

1Step 1: Understand the Mean Value Theorem

The Mean Value Theorem states that for a continuous function on a closed interval $[a,b]$ and differentiable on the open interval $ (a, b) $, there exists at least one point $ c $ in $(a, b)$ such that $rac{f(b) - f(a)}{b - a} = f'(c)$. This results in the conclusion that $ f(b) = f(a) + f'(c)(b-a) $.

2Step 2: Setup the Tangent Line Equation

For any point $ c $ in the interval, the equation of the tangent line to $ f $ at $ x = c $ is given by $ f(x) = f(c) + f'(c)(x-c) $. This is a linear approximation of $ f $ at $ x = c $.

3Step 3: Define Concavity

A function is concave up if its second derivative is positive $ f''(x) > 0 $. This implies that the slope of the tangent increases as you move away from any given point within the interval. Therefore, the graph of the function is curving upwards.

4Step 4: Compare Function and Tangent Line Values

Since $ f $ is concave up (i.e., $ f''(x) > 0 $), the function will always lie above its tangent line at any $ x eq c $. This is because for any small changes in $ x $, $ f(x) $ increases more rapidly than the tangent line does.

Key Concepts

Concave Up FunctionTangent LineSecond DerivativeDifferentiable Function

Concave Up Function

A function is described as 'concave up' if the graph of the function bends upwards like a cup. This property is mathematically captured by the second derivative of the function, which needs to be positive over its domain:

If $ f''(x) > 0 $, the function $ f(x) $ is concave up.
This indicates that the slope of the tangent line increases as you move along the x-axis.

With a concave up function, the behavior is such that as you go from left to right, the graph does not ever decrease or level out; instead, it maintains a constant upward bending shape.
As a result of this consistent upward bend, the graph of a concave up function will, in fact, lie above its tangent line at any point except at the point of tangency. This is pivotal when using the Mean Value Theorem to demonstrate the above behavior.

Tangent Line

The tangent line to a curve at a given point is the straight line that touches the function at that point and has the same slope as the function does at that point. Geometrically, it is the line that just "kisses" the curve.

For a function $ f(x) $ at a point $ c $, the equation of the tangent line is $ f(x) = f(c) + f'(c)(x-c) $.
The slope of the tangent line is the first derivative of the function at that point, $ f'(c) $.

Understanding the tangent line helps in approximating the function near $ c $, but in the context of a concave up function, it also helps illustrate that the function curve will, indeed, lie above this tangent.

Second Derivative

The second derivative of a function provides important insights into the function's concavity. It is essentially the derivative of the derivative, revealing how the rate of change of the function itself is changing.

If $ f''(x) > 0 $, the function is concave up, indicating that the slope is increasing.
If $ f''(x) < 0 $, the function would be concave down.

The utility of the second derivative in the Mean Value Theorem and in verifying the position of a concave up function in relation to its tangent line is immense. A positive second derivative confirms that the slope of the tangent is increasing, ensuring that the graph of the function remains above its tangent line outside the specific point of tangency, an essential part of demonstrating the function's behavior.

Differentiable Function

A differentiable function is a function that has a derivative at each crucial point in its domain. Differentiability implies that the function has a well-defined tangent at each point, making the entire concept of tangents and the Mean Value Theorem applicable.

For a function to be differentiable at a given point $ c $, it must be smooth (no sharp edges or cusps) and continuous at that point.
This ensures that the tangent line can be calculated and that the linear approximation holds.

The requirement that a function is differentiable on an interval is one of the key conditions of the Mean Value Theorem. Without differentiability, we can't guarantee that a tangent exists, hindering our ability to compare the graph of the function to its tangent line effectively.

Problem 48

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