Given information: f(x)=x3−4x+1

Theorem used:

Intermediate Value Theorem: If f is a continuous function on [a,b] and if y is a number between f(a) and f(b), then there exists a number c with a≤c≤b such that f(c)=y .

Calculation:

Consider the equation, f(x)=x3−4x+1

Substituting x=1 in f(x)=x3−4x+1 , we get

  f(1)=13−4(1)+1=1−4+1=−2

Substituting x=0 in f(x)=x3−4x+1 , we get

  f(0)=03−4(0)+1=0−0+1=1

Therefore, we have

  −2<0<1f(1)<0<f(0)

Using Intermediate value Theorem, there exists c∈(0,1) such that f(c)=0 .

Hence, it is proved.

Given information: f(x)=5cosπx−4

Theorem used:

Intermediate Value Theorem: If f is a continuous function on [a,b] and if y is a number between f(a) and f(b), then there exists a number c with a≤c≤b such that f(c)=y .

Calculation:

Consider the equation, f(x)=5cosπx−4

Substituting x=1 in f(x)=5cosπx−4 , we get

  f(1)=5cosπ−4=5(−1)−4=−5−4=−9

Substituting x=0 in f(x)=5cosπx−4 , we get

  f(0)=5cosπ(0)−4=5−4=1

Therefore, we have

  −9<0<1f(1)<0<f(0)

Using Intermediate value Theorem, there exists c∈(0,1) such that f(c)=0

Hence, it is proved.

Given information: f(x)=8x4−8x2+1

Theorem used:

Intermediate Value Theorem: If f is a continuous function on [a,b] and if y is a number between f(a) and f(b), then there exists a number c with a≤c≤b such that f(c)=y .

Calculation:

Consider the equation, f(x)=8x4−8x2+1

Substituting x=1 in f(x)=8x4−8x2+1 , we get

  f(1)=8(1)4−8(1)2+1=8−8+1=1

Substituting x=0 in f(x)=8x4−8x2+1 , we get

  f(0)=8(0)4−8(0)2+1=0−0+1=1

Substituting x = 12 in f(x)=8x4−8x2+1, we get

  f(12)=8( 1 2)4−8( 1 2)2+1=8(1 16)−8(14)+1=12−2+1=−12

Therefore, we have

  −12<0<1f(12)<0<f(0)

Using Intermediate value Theorem, there exists c∈(0,12) such that f(c)=0 .

Hence, it is proved.