Problem 19 - Mathematics (Differential Calculus): Find the value of x, prescribed by the law of the mean given f(x) = 3x² + 4x - 3, a = 1, b = 3.

The answer

The rule

If a function f(x) is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that: f'(c) = [f(b) - f(a)] / (b - a)

Formula: f'(c) = [f(b) - f(a)] / (b - a)

Let's understand it simply

Imagine you are riding your cycle from your home (at x = 1) to the market (at x = 3). Your speed changes as you go. The Mean Value Theorem (MVT) says: at some point, your actual speed will be exactly equal to your average speed for the whole trip. Here, f(x) is like your position at time x.

How it works

1. Check if f(x) is continuous and differentiable between 1 and 3. (It is, since it is a polynomial.)
2. Find f(a) and f(b):

• f(1) = 3(1)² + 4(1) - 3 = 3 + 4 - 3 = 4
• f(3) = 3(3)² + 4(3) - 3 = 27 + 12 - 3 = 36

1. Find the average rate of change:

• [f(3) - f(1)] / (3 - 1) = (36 - 4) / 2 = 32 / 2 = 16

1. Find f'(x):

• f'(x) = d/dx [3x² + 4x - 3] = 6x + 4

1. Set f'(x) = 16 and solve for x:

• 6x + 4 = 16
• 6x = 12
• x = 2

1. Check if x = 2 is in (1, 3). Yes, it is.

Worked example

Given: f(x) = 3x² + 4x - 3, a = 1, b = 3
Step 1: f(1) = 3(1)² + 4(1) - 3 = 4
Step 2: f(3) = 3(3)² + 4(3) - 3 = 36
Step 3: Average rate = (36 - 4) / (3 - 1) = 16
Step 4: f'(x) = 6x + 4
Step 5: 6x + 4 = 16 → 6x = 12 → x = 2

Why each step works

• Step 1 & 2: Find the function's values at the endpoints.
• Step 3: Calculate the average rate of change (like average speed).
• Step 4: Find the derivative (instantaneous rate).
• Step 5: Set the derivative equal to the average rate and solve for x.

Where you see it in real life

• Driving from one city to another: At some point, your speed matches your average speed.
• Filling a water tank: At some time, the rate you are filling matches the average rate.
• Cricket: A batsman scores runs at different rates, but at some ball, his scoring rate matches the average.
• Business: The rate of sales on a day matches the average sales rate at some hour.

Why it matters

• Used in physics to relate average and instantaneous speed.
• Helps in understanding how things change over time.
• Used in economics to analyze growth rates.
• Important for higher maths and engineering.

Common mistakes

• Forgetting to check if the function is continuous and differentiable.
• Using endpoints (a or b) instead of a value strictly between them.
• Not setting the derivative equal to the average rate.

Remember it

Mean Value Theorem: At some point, the slope of the curve equals the average slope between the endpoints.