Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem. (Enter your answers as a comma-separated list.)
f(x) = x3 − x2 − 20x + 3, [0, 5] c =
1) Function is continuous on the interval [a,b]
2) Function is differentiable on the interval (a,b)(
3) Function satisfies f (a) = f (b)
Given function is a polynomial which means that it is continuous and differentiable on R.
This gives us 1) and 2) because if a function is continuous and differentiable on R, then it is continuous and differentiable on each interval in R, including [0,5].
As for the third part:
So, all three hypotheses of Rolle’s Theorem are satisfied.
Now, find derivative using rules of derivatives and find all points where derivative is zero.
So, function has derivative zero at points