Prove that the following functions do not have maxima or minima:
f(x) = eˣ
(i) f(x) = eˣ f'(x) = eˣ For maxima or minima, f'(x) = 0. eˣ = 0 has no solution since eˣ > 0 for all real x. Hence, f(x) = eˣ has no maxima or minima.
g(x) = log x
(ii) g(x) = log x g'(x) = 1/x For maxima or minima, g'(x) = 0. 1/x = 0 has no solution. Hence, g(x) = log x has no maxima or minima.
h(x) = x³ + x² + x + 1
(iii) h(x) = x³ + x² + x + 1 h'(x) = 3x² + 2x + 1 For maxima or minima, h'(x) = 0. 3x² + 2x + 1 = 0 Discriminant = b² - 4ac = 4 - 12 = -8 < 0 Since discriminant < 0, the equation has no real roots. Hence, h(x) has no maxima or minima.
Explanation
To prove a function has no maxima or minima, we find the derivative and check if f'(x) = 0 has any real solutions. If no critical points exist (where derivative equals zero), the function cannot have local maxima or minima. For eˣ, the derivative eˣ is always positive. For log x, the derivative 1/x never equals zero. For the cubic polynomial, the discriminant of the derivative equation is negative, indicating no real roots.