MEAN THEOREM

--    Verify Rolle's theorem --

1) 

a) f(x)= x(x-4)²on (0,4).           

b) f(x)= x²-4x+3 on (1,3).                

c) f(x)= x(x-2)² on (0,2).

d) f(x)= x²+ 5x+6 on (-3,-2)

e) f(x)= x² - 8x+6 on (2,6)

f) f(x)= (x-1)(x-2)² on (1, 2).

g) f(x)= x(x²-1)² on (0,1).

h) f(x)= (x²-1)(x-1) on (-1,2).

i) f(x)= eˣ Sin x in (0,π).      

j) f(x)= cos 2(x - π/4) on (0,π/2).

k) f(x)= sin2x on (0,π/2).

l) f(x)= cos2x on (-π/4,π/4).

m) f(x)= eˣ cosx in (-π/2,π/2).  

n) f(x)=  Sin x /eˣ on 0≤ x ≤π  

o) f(x)= sin 3x on (0,π).

p) f(x)=sinx + cosx on (0,π/2).

q) f(x)=2 sinx + sin2x on (0,π).

r) f(x)=6x/π - 4 sin²x on (0,π/6)

s) f(x)= sin²x on 0 ≤ x ≤π.

t) f(x)= sinx + cosx -1 on (0,π/2)

u) f(x)= log(x²+2) - log3 on (-1,1)

2) At what points on the following curves, is the tangent parallel to x-axis?

a) y= 16 - x² on (-1,1).               (0,16)

b) y= x² on (-2,2).                      (0,0)

c) y= 12(x+1)(x-2) on (-1,2).  (1/2,-27).

d) y= cosx -1, on (π/2, 3π/2).   (π,-2)

3) It is given that for the function f(x)= x³ - 6x³ + ax + b on (1,3), Rolle's theorem holds with c= 2 + 1/√3. Find the values of a and b, if f(1)= f(3)= 0.                              11, -6

4) It is given that for the function f given by f(x)= x³ + bx²+ ax, on (1,3). Rolle's theorem holds with c= 2+ 1/√3. Find the values of a and b.  11,-6

-- Verify Lagrange's mean value --

1)

a) f(x)= x(x -2) on (1,3).

b) f(x)= x² -1 on (2,3).

c) f(x)= x³- 2x² -x+3 on (0,1).

d) f(x)= x² -3x+2 on (-1,2).

e) f(x)= 2x² -3x+1 on (1,3).

f) f(x)= x² -2x+4 on (1,5).

g) f(x)= 2x- x²  on (0,1).

h) f(x)= (x -1)(x-2)(x-3) on (0,4).

I) f(x)= √(25-x²) on (-3,4).

j) f(x)= x +1/x on (1,3).

k) f(x)= x(x+4)² on (0,4).

l) f(x)= √(x² -4) on (2,4).

m) f(x)= x²+ x -1 on (0,4).

n) f(x)=sin x -sin2x on (0,π).

o) f(x)= tan⁻¹x on (0,1).

p) f(x)= log x on (1,2).

2) Find a point on the parabola y= (x-4)², where the tangent is parallel to the chord joining (4,0) and (5,1).    (9/2,1/4)

3) Find a point on the curve y= x²+x, where the tangent is parallel to the chord joining (0,0) and (1,2).    (1/2,3/4)

4) Find a point on the parabola y= (x- 3)², where the tangent is parallel to the chord joining (3,0) and (4,1).    (7/2,1/4)

5) Find the points on the parabola y= (x³ - 3x, where the tangent to the curve parallel to the chord joining (1,-2),(2,2).    (±√(7/3, ±2/3 √(7/3), 

6) Find a point on the curve y= x³+1 where the tangent is parallel to the chord joining (1,2),(3,28).   (√(13/3, ²√13/3)³ +1.