I find that the harder I work, the more luck I seem to have. – Thomas Jefferson

Topics discussed in this lecture:

Meaning of Local maxima, local minima, local maximum and local minimum
First derivative test for local maxima and local minima

NCERT EXERCISE 6.5

Questions discussed in this lecture

Question 3. Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
(i) \( f(x) = x^2 \)
(ii) \( g(x) = x^3 – 3x \)
(iii) \( h(x) = \sin x + \cos x , 0<x<\frac{\pi}{2} \)
(iv) \( f(x) = \sin x – \cos x , 0<x<2\pi \)

(v) \( f(x) = x^3 – 6x^2 + 9x + 15 \)

(vi) \( g(x) = \frac{x}{2} + \frac{2}{x},  x>0 \)
(vii) \( g(x) = \frac{1}{x^2 + 2} \)
(viii) \( f(x) = x \sqrt{1 – x}, 0<x<1 \)

Question 4. Prove that the following functions do not have maxima or minima:
(i) \( f(x) = e^x \)
(ii) \( g(x) = \log x \)
(iii) \( h(x) = x^3 + x^2 + x + 1 \)