MediumCalculusca-038

Mean Value Theorem Point

Problem

Find the smallest point c in [a, b] guaranteed by the Mean Value Theorem for the polynomial f(x) = sum(coeffs[i] * x^i), so that f'(c) equals the average slope (f(b) - f(a)) / (b - a).

Solve the quadratic f'(c) - slope = 0 exactly and return the smallest valid root, or None if no root lies in the interval.

Hint

Set f'(c) equal to the secant slope and solve the resulting polynomial equation.

Test cases

case 1

in: [[0,-3,0,1],0,2]

expected: 1.1547005383792517

case 2

in: [[0,0,0,1],0,3]

expected: 1.7320508075688772

case 3

in: [[0,1,0,1],0,1]

expected: 0.5773502691896258

case 4

in: [[0,-3,0,1],-2,2]

expected: -1.1547005383792517

case 5

in: [[0,0,1],0,2]

expected: 1

Starter codePython

def mvt_point(coeffs, a, b):
    # Return the smallest valid c, or None
    # Your code here
    pass

5 test cases · run your code against them in the editor