Identify this

Algebra Level 4

Given that $f$ is a real valued differentiable function such that $\large{f(x)f^{\prime}(x)<0}$ $\forall~real~x$ . Then it follows that

Details

$abs(f(x))=|f(x)|$

$f^{\prime}(x)=\frac{d }{dx}f(x)$

abs(f(x))

is an increasing function

f(x)

is an increasing function

f(x)

is a decreasing function

abs(f(x))

is an decreasing function

1 solution

Tijmen Veltman
Apr 28, 2015

We have:

$sgn\left(\frac{d}{dx}abs(f(x))\right) =sgn\left(sgn(f(x))\cdot f'(x)\right) =sgn\left(f(x)f'(x)\right)=-1.$

Hence $\frac{d}{dx}abs(f(x))<0$ for all $x\in\mathbb{R}$ , meaning that $abs(f(x))$ is a decreasing function.

so wrong sum.......what if the function is of sort e^-x,huh???then f(x) is decreasing !and it satisfies the inequality...( nice solution though) cmon u have to accept the sum is fishy!

Rushikesh Joshi - 6 years ago

$f(x)=e^{-x}$ is indeed a function that satisfies the conditions. We have $f(x)f'(x)=-e^{-2x}<0$ and $abs(f(x))=e^{-x}$ is a decreasing function.

Tijmen Veltman - 6 years ago

Identify this

1 solution

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