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# Exercise 7.7: Applications of Second Derivative

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EXERCISE 7.7

1. Find intervals of concavity and points of inflexion for the following functions:

(i) f (x= x ( x ŌłÆ 4)3

(ii) f (x= sin x + cos x, 0 < x < 2ŽĆ

(iii) f (x) = 1/2 ( ex ŌłÆ eŌłÆx )   2. Find the local extrema for the following functions using second derivative test :

(i) f (x= ŌłÆ3x5 + 5x3

(ii) f (x= x log x

(iii) f (x= x2 eŌłÆ2x 3. For the function f (x= 4x3 + 3x2 ŌłÆ 6x +1 find the intervals of monotonicity, local extrema, intervals of concavity and points of inflection. (1) (i) concave upwards on ( ŌłÆŌł×, 2) and ( 4, Ōł×) . Concave downwards on ( 2, 4)

Points of inflection ( 2, ŌłÆ16) and ( 4, 0)

(ii) concave upwards on . Concave downwards on Points of inflection (iii) concave upwards on ( 0, Ōł×). Concave downward on ( ŌłÆŌł×, 0)  Points of inflection ( 0, 0)

(2) (i) local minimum = ŌłÆ2 ; local maximum = 2 (ii) local minimum = ŌłÆ 1/e  (iii) local minimum = 0 ; local maximum = 1/e2

(3) strictly increasing on (ŌłÆŌł×, ŌłÆ1) and (1/2 , Ōł×) . strictly increasing on (ŌłÆ1, 1/2) local maximum = 6 , local minimum = ŌłÆ 3/4  concave downwards on (ŌłÆŌł×, ŌłÆ1/4) ; concave upwards on (ŌłÆ 1/4, Ōł×) .  point of inflection (ŌłÆ 1/4, 21 /8)

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12th Maths : UNIT 7 : Applications of Differential Calculus : Exercise 7.7: Applications of Second Derivative | Problem Questions with Answer, Solution