Function Property

Calculus Level 2

If a function $f(x)$ that is differentiable over $(-\infty,\infty)$ is monotonically decreasing and $\displaystyle\lim_{x\rightarrow\infty}f(x)\neq-\infty,$ then as $x$ approaches infinity, $f(x)$ is

Increasing Concave up Impossible to determine Equal to 0 or

+\infty

Concave down

1 solution

Basim Khajwal
Mar 14, 2017

Using the fact that the value of $f(x)$ does not approach negative infinity as $x$ approaches infinity it must be the case that the value of $f(x)$ converges to some value $k$ .

Since the function is monotonically decreasing the gradient must be negative but, in order to converge, the gradient of the function must approach $0$ towards infinity. Hence, the second derivative must be positive therefore the function is concave up.

Could anyone explain Pls I'm new

Tarun B - 3 years, 2 months ago

Well, there is an irreparable gap in this reasoning: true, the derivative ("gradient") is negative and true, it must approach 0. But that's all we know, that doesn't imply that the derivative must be monotonly approaching 0, it can for instance go like say -(sin(x)/x)^2

J T - 2 years ago

It's concave as seen from right, not from above!

A Former Brilliant Member - 2 years, 4 months ago

Attention! The solution given by the poster is wrong, see report for counterexamples and explanation.

J T - 1 year, 9 months ago

Function Property

1 solution

1 pending report

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