Verify that each function satisfies the hypothesis of the Mean Value theorem on the interval [a,b] and find all the numbers c that satisfy the conclusion..
.. Of the Mean Value Theorem
G(x) =x^3 + x - 4 on [-1,2]
The function is continuous on [-1,2] and differentiable on (-1,2) so the function satisfies the hypothesis for the mean value theorem.
The mean value theorem (MVT) states that there exists at least one c in [-1,2] such that `f'(c)=(f(2)-f(-1))/(2-(-1))` .
`f'(x)=3x^2+1` Setting f'(x)=4 we get:
So the values of x where the slope of the tangent line to the function is equal to the average rate of change of the function over this interval are -1 and 1.
The graph of the function, the secant line, and the tangent lines:
Note that one of the tangent lines is the secant line.