Maths Doubt

[The Villain]

F(x,y) = y lnxy -3 then gradient vector at (1/2,2) is

a) (16,1)

b) (4,1)

c) (4,2)

d) (16,2)

[Noel]

Is it b?

[The Villain]

Yeah its b...is it (fx,fy)??

[Noel]

Yup

[The Villain]

Thanxx...can u help me with this one..

The function dened by f(x) = x5 + 7x3 + 13x- 18
(a) may have 5 real roots.
(b) has no real root.
(c) has 3 real roots.
(d) has exactly 1 real root.

[Noel]

Is it d?

[The Villain]

I don't know the answer man.How did you solve it?
Is there any specific method to solve such questions.Please help..

[Noel]

f has a root in (0,1) because f(0) is negative and f(1) is positive and f being a polynomial is continuous

now f '(x)=5x^4 +21x^2 +13 which is always +ve so since f is always increasing hence there can be only one real root

[The Villain]

Thanxxx man
so for such questions we need to check in this manner??

[Noel]

Well..yes though it is needless to say that every question of this sort may not have a very similar approach to this