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

1.	Prove that the following functions do not have maxima or minima : (i) f (x) = ex(ii) g(x) = log x, x > 0(iii) h (x) = x3 + x1 + x +1
Answer» (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.

Prove that the following functions do not have maxima or minima : (i) f (x) = ex(ii) g(x) = log x, x > 0(iii) h (x) = x3 + x1 + x +1

Answer»

(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.