Question

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

Answer 1

i. We have, f(x) = ex

∴f'(x)=ex

Now, if f'(x)=0,then ex=0. But, the exponential function can never assume 0 for any value of x.

Therefore, there does not exist c∴ R such that f'(c)=0

Hence, function f does not have maxima or minima.

possitive numbers x, g'(x)>0

Therefore, ther=g'(c)=0 g does not exist c∴ R such that g(x) = log x.

Hence, function g does not have maxima or minima.

iii. We have,

h'(x) = x³+x²+x+1

h'(x)=3x²+2x+1

Now,

h(x) = 0 ∴ 3x²+2x+1 = 0 ∴x=-2±2\(\sqrt{\frac{2i}{6}}\)=-1±\(\sqrt{\frac{2i}{3}}\)∉R

Therefore, there does not exist c∴ R such that h'(c)=0.

Hence, function h does not have maxima or minima