Class: 12 Subject: Math Type: Long

Question 1: Show that the function given by \( f(x) = \dfrac{\log x}{x} \) has maximum at \( x = e \)....

Class: 12 Subject: Math Type: Long

Question 2: Find the intervals in which the function \( f(x) = \dfrac{4\sin x - 2x - x\cos x}{2 + \cos x} \) is (i) increasing (ii) decreasing....

Class: 12 Subject: Math Type: Long

Question 3: Find the intervals in which the function \( f(x) = x^3 + \dfrac{1}{x^3}, x \ne 0 \) is (i) increasing (ii) decreasing....

Class: 12 Subject: Math Type: Long

Question 4: A point of the hypotenuse of a triangle is at distance \( a \) and \( b \) from the sides of the triangle. Show that the minimum length of the hypotenuse is \( \left( a^{2/3} + b^{2/3} \right)^{3/2} \)....

Class: 12 Subject: Math Type: Long

Question 5: Find the points at which the function \( f(x) = (x - 2)^4(x + 1)^3 \) has (i) local maxima (ii) local minima (iii) point of inflection....