1. Prove that there exists a point ‘c’ in the interval [-1,3] for which function f:R->R continuous and differentiable in [0,2] such that f(0) = f(2) = 7 satisfies f’(c) = 0. Also find this ‘c’ for f(x) = x2 in interval [-1,1].

2. Show that the equation f’(x) = 0 has at least one real root in the interval [0,2] where f(x) = x4 – 4x3 + 8x + 33.

3. Let f be a function continuous on [a,b], can we say that if f(a) = f(b), then Ǝ (there exist) c€(a,b) such that f’(c) = 0.

4. At least how many horizontal tangents, will the function f(x) = (x-1)(x-2)(x-3)……..(x-n) in the interval (0,n)?

5. Prove that Ǝ (there exist) c€[a,b] such that f’(c)g(c) – g’(c)f(c) = (g(c))2 if f(a) + 2f(b) = g(a) + 2g(b) = 0.

6. Prove that f’(x) = 0 if and only if f(x) is a constant function.

7. Prove that 1 – cos x ≤ x2/2 for x€[0,π/2).

8. At least how many tangents inclined at an angle tan-1(2) for the function f(x) = x2.
9. Use MVT to prove that |tan x – tan y|≥|x - y|.

10. Let f:R->R be doubly differentiable such that f(0) = f(2) = 7, f(3) = f(4) = 9.
Prove that Ǝ c€(0,4) such that f”(c)=0.

11. Let f be doubly differentiable such that f(1) = 7, f(2) = 9, f(32) = 61, f(33) = 63, prove that Ǝ α € (1,33) such that f”(α) = 0 .

12. Find the number of roots of x2 = xsin x + cos x.

13. Prove that log(1+x) > x/(1+x) for x>0 Hence show that log(1+x) > log(x) + x/(1+x).

14. Show that cos x < 1 – x2/2! + x4/4!

15. Prove that (1+x)α > 1+αx for x>1.

16. If af(a) = bf(b), where f is continuous and differentiable function in [a,b] , prove that αf’(α) + f(α) = 0, for some α € (a,b).

17. If f is continuous and differentiable in (a,b), then f satisfies rolle’s theorem, true or false ?

18. If f, g, h be three functions such that f(a)g(a)h(a) – f(b)g(b)h(b) = a – b.
Prove that Ǝ α € (a,b) such that f’(α)/f(α) + g’(α)/g(α) + h’(α)/h(α) = (f(α)g(α)h(α))-1

19. If a function f has local maxima or minima at a point, then tangent at that point must be horizontal, true or false?

20. Apply rolle’s theorem on a function g, continuous in interval [a,b] and differentiable in (a,b) with g(x) = f(x) – (f(b) – f(a))(x-a)/(b-a).

21. Can local maxima and minima occur simultaneously? Give example.

22. The minimum of all local minimum of all local minimum over an interval is called absolute minimum over that interval? (True or false)