Question

Let f(x) be a differentiable function and G be the graph of f(x).Let P={a,f(a)} be a point on G closest to (0,0).Then f(a)f'(a)=?

Answer 1

@ganeshie8

Answer 2

use the distance formula and minimise distance between a generalised point (x, f) [where f = f(x)] and the Origin by using \(s^2 = x^2 + f^2\)

minimising \(\implies 2 s s' = 2x + 2f f' = 0\) at (a, f(a))

see it?

if the line goes through the origin, you have a problem, i think, but not sure you need to worry about that

Answer 3

Actually i did this problem considering this.
Slope of tangent at point A is f(a)/a (since it is close to origin so it must pass through origin).
Now question says f(a)f'(a) so we can substitute it here.
THAT BECOMES f(a)^2/a.
Now , i assume that f(a) is a only.
So got the ans to be a.

Answer 4

i think it's -a

FWIW

Answer 5

Your ans is correct.
Will you please tell how did get it?

Answer 6

You*

Answer 7

it's simply a minimisation problem. find the distance in terms of \(x\), and then minimise the function. \[d(x) = \sqrt{x^2+y^2}=\sqrt{x^2 + f^2(x)}\]now we have to minimise the above expression. we can also say that we're minimising \(g(x) = x^2 + f^2(x)\). that will be true when \(g'(x) = 0\).

now \(g'(x) = 2x + 2f(x)f'(x) = 0\). we're given that this equation holds at \(x = a\). this means\[2a + 2f(a)f'(a) = 0 \Rightarrow f(a)f'(a) = -a \]