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) \( h(x) = x^3 + x^2 + x + 1 \)