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

Solution:

Maxima and minima are known as the extrema of a function.

Maxima and minima are the maximum or the minimum value of a function within the given set of ranges.

According to the given question.

(i) f (x) = e^x

f' (x) = e^x

Now,

f' (x) = 0

⇒ e^x = 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.

(ii) g (x) = log x

Therefore,

g' (x) = 1/x

Since log x is defined for a positive number x, g' (x) > 0 for any x.

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

Hence, function g does not have maxima or minima.

(iii) h (x) = x³ + x² + x + 1

Therefore,

h' (x) = 3x² + 2x + 1

Now,

h' (x) = 0

⇒ 3x² + 2x + 1 = 0

⇒ x = (- 2 ± √2i) / 6

⇒ x = (- 1 ± √2i) / 3 ≠ R.

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

Hence, function h does not have maxima or minima

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.5 Question 4

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

Summary:

i)There does not exist c ∈ R such that f' (c) = 0.Hence, function f does not have maxima or minima. Maxima and minima are the maximum or the minimum value of a function within the given set of ranges