Find the value c in [-1,2] that satisfies the Mean Value Theorem.

Question

Consider the function f(x)= x³+ x on [-1,2]. Find the value c in [-1,2] that satisfies the Mean Value Theorem.

Answer 1

f(x) = x^3 +x on [-1,2] find c that satisfies MVT

f(2) = 10, f(-1) = -2

(f(2) - f(-1))/(2+1) = (10+2)/3 = 4 = slope of the secant line on the interval

f'(c) = 4 = 3x^2 +1

3x^2 =3

x^2 =1

x = 1 = c

f'(c) = f'(1) = 4 = 3(1)^2 + 1

at x=1 the instantaneous rate of change = average rate of change. on the given interval

slope of tangent line = slope of secant line

c = 1

as a rough check on the solution, use a graphing calculator and graph f(x)

visually estimate the secant line slope on the interval and the tangent line slope at x=1

the function is continuous on the closed interval

and differentiable on the open interval

satisfying the Mean Value Theorem