Question

Prove that the following functions do not have maxima or minima : 

(i) f (x) = e^x

(ii) g(x) = log x, x > 0

(iii) h (x) = x³ + x¹ + x +1

Answer 1

(i) f(x) = e^x 

f’(x) = e^x 

f'(x) > 0 ∀ x ∈ R 

hence function has no critical point There is no point at which the function is maximum or minimum. 

(ii) g(x) = log x, x > 0 

g'(x) = \(\frac{1}{x}\), where x > 0 

hence the function has no critical point 

∴ There is no point at which the function is maximum or minimum. 

(iii) h(x) = x³+ x²+ x +1 

h’(x) = 3x² + 2x + 1 

h’(x) = 0 ⇒ 3x² + 2x + 1 = 0, 

x has no real value, hence there is no critical point. 

∴ For no point the function has max. or min. value.