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**Question 16: [2017 : 1 Mark, Set-I]****Solution: **(Applying L'Hospital rule)

= **Question 17: The quadratic approximation of ****f(x) = x ^{3} - 3x^{2} - 5 a the point x = 0 is [2016 : 2 Marks, Set-II]**

and straight is y = 0

At the point of intersection, we have,

â‡’

x

and y

y = x

x

â‡’ x = -2, 1 and 3 - x

(Using definition of Laplace transform)

Put s - 0, we get

(i.e., put x = 0 and then y = 0)

which depends on m.

f'(x) = 0

â‡’ 2x â€” 4 = 0

â‡’ x = 2 (stationary point)

f"(x) = 2 > 0

â‡’ f(x) is minimum at x = ?

i.e., (2)^{2} - 4(2) + 2 = -2

âˆ´ The optimum value of f(x) is -2 (minimum)**Question 24: While minimizing the function f(x), necessary and sufficient conditions for a point x _{0} to be a minima are [2015 : 1 Mark, Set-II]**

if f'(x

Which is in the form of

To convert this into 0/0 form, we rewrite as,

Now it is in 0/0 form.

Using Lâ€™Hospitalâ€™s rule,

âˆ´ y = e

Directional derivative

**â‡’ ****Alternative Method:****Question 30: For the parallelogram OPQR shown in the sketch, The area of the parallelogram is [2011 : 2 Marks]****(a) ad - bc ****(b) ac + bd ****(c) ad + bc ****(d) ab - cd****Answer: **(a)

The area of parallelogram OPQR in figure shown above, is the magnitude of the vector product

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