Answer to Question #288626 in Calculus for Nikk

Question

The function f defined by f(x) = tan(2x) is a periodic function with period π. ii) The function 𝑓:𝑹 → 𝑹, defined by 𝑓(𝑥) = 1 − |𝑥| is differentiable at x=1. iii) The function 𝑓: [3,4] → 𝑹 defined by 𝑓(𝑥) = 𝑥 ଶ − 𝑥 is monotonic in its domain. iv) Every continuous function is differentiable. v) Every curve over R has a point of inflection. 

Answer 1

(i)

Function f(x) = tanx is a periodic function with period $\pi$.

So, period of tan2x is $\pi$/2. This is because period of tanax is $\pi$/a.

The given statement is false.

(ii)

Lets graph f(x) = 1 - |x|

Clearly, at x = 1, there is only tangent. So, the function is differentiable at x = 1.

The given statement is true.

(iii)

$f(x)=x^2-x$

A monotonic function is a function which is either entirely nonincreasing or nondecreasing.

The given function is nondecreasing in [3,4], so it is monotonic in its domain.

So the given statement is correct.

iv)

for example:

$f(x)=|x|$ is continuous but not differentiable at x = 0.

statement is false

v)

if the second derivative can equal zero, the original function has a point of inflection

for example:

$y=e^x$

$y''=e^x$ cannot equal zero, so the curve has not a point of inflection

statement is false