# math

posted by .

Suppose that function f:[0,2]->R is continuous at [0,2], differentiable at (0,2) and such that f(0)=0, f(1)=1 and f(2)=1.
a) Show that there is a x1 that belongs to (0,1) such that f'(x1)=1.
b) Show that there is a x2 that belongs to (1,2) such that f'(x2)=0.
c) Show that there is a x3 that belongs to (0,2) such that f'(x3)=2/3.

• math -

a) (f(1)-f(0))/(1-0) = 1
MVT says x1 exists

b) f(1)=f(2), so MVT or Rolle's Theorem says x2 exists

c) since differentiable, the IVT applies to the derivative, which must assume all values between 0 and 1 (which we know it attains from the two parts above).

## Similar Questions

1. ### calculus

In the following problem, suppose f(x) is continuous (and differentiable) function on the interval (0,1). Suppose also that for 0 < x<(1/4) f(x) is negative, for (1/4) <x<1 f(x) is positive, f(1/4)=0, f (2/3)=1, f ' (1/3)= …
2. ### Calculus

Let f be a twice-differentiable function such that f(2)=5 and f(5)=2. Let g be the function given by g(x)= f(f(x)). (a) Explain why there must be a value c for 2 < c < 5 such that f'(c) = -1. (b) Show that g' (2) = g' (5). Use …
3. ### Calculus I

Suppose that f and g are two functions both continuous on the interval [a, b], and such that f(a) = g(b) = p and f(b) = g(a) = q where p does not equal to q. Sketch typical graphs of two such functions . Then apply the intermediate …
4. ### calc

Which of the following statements would always be true?
5. ### calculus

let f:[0,1] - [0,1] be continuous function .show that there exist a in [0,1] such that f(a)=a
6. ### Math11

Hello, I don't know how to do this, please help. Thank you. 1).Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval?
7. ### Calculus

(3) Suppose that f : R2 → R is a continuous function at x0 ∈ R2 and that |f(x0)| > 2. Show that there is a \$ > 0 such that |f(x)| > 2 whenever ||x − x0|| < \$.
8. ### calculus

a) Obtain all solutions of the equation z^3 +1 = 0 b) Let z = x + iy. Obtain the real and imaginary parts of the function f(z) = 1/1+z c) Let f(x + iy) = x^2 - y^2 + iv( x,y). Determine a function v such that f is differentiable in …
9. ### Math

Is the step function differentiable in its domain?
10. ### Math - Calculus

Show that if f(x) is continuous and u(x) and v(x) are differentiable functions, then: d/dx ∫_v(x)^u(x) f(t)dt=u'(x)f(u(x))-v'(x)f(v(x))

More Similar Questions