Question

4.Prove that the following functions do not have maxima or minima:(i) f(x)=ex(iii) h(x)=x3 +x2 + x +1(ii) g(x)=log x

Answer 1

(i)

Given function is,

f( x )= e x

Differentiate the function with respect to x,

f ′ ( x )= e x

Put f ′ ( x )=0,

e x =0

It is not possible for any value of x.

Therefore, f( x )= e x have neither maxima nor minima.

(ii)

Given function is,

g( x )=logx

Differentiate the function with respect to x,

g ′ ( x )= 1 x

Put g ′ ( x )=0,

1 x =0 x= 1 0 x=∞

The point x=∞ is not defined for the given function.

Therefore, g( x )=logx, have neither maxima nor minima.

(iii)

Given function is,

h( x )= x 3 + x 2 +x+1

Differentiate the function with respect to x,

h ′ ( x )=3 x 2 +2x+1

Put h ′ ( x )=0,

3 x 2 +2x+1=0 x= −2±2 2 i 6 x= −1± 2 i 3

There does not exist any real value.

Therefore, h( x )= x 3 + x 2 +x+1 have neither maxima nor minima.