Answer to Question #103917 in Calculus for Martin

Question

Is the graph of y=sin(x5) increasing or decreasing when x=13?
(enter increasing, decreasing, or neither).

Is it concave up or concave down?

Answer 1

let's find a first derivative

$y'=(sin(x^5))'=(sin(x^5))'*(x^5)'=5*x^4*cos(x^5)$

now we have to understand whether the graph y' increasing or decreasing. So let's find the value of the derivative y' at x=13

$y'=5*x^4*cos(x^5)=5*13^4*cos(13^5)=106353\implies y'>0$ ,

so the graph is increasing at x=13.

Next question - is the graph concave up or concave down. Let's find second derivative y"

$y''=(sin(x^5))''=(5*x^4*cos(x^5))'=(x^4)'*cos(x^5)+x^4*(cos(x^5))'=4*x^3*cos(x^5)+x^4*(-5*x^4*sin(x^5))=-25*x^8*sin(x^5)+20*x^3*cos(x^5)$

to understand is the graph concave up or concave down let's find y'' at x=13

$y''=-25*x^8*sin(x^5)+20*x^3*cos(x^5)=5*x^3*(4*cos(x^5)-5*x^3*sin(x^5))$

$y''= -13609384564.031\implies y''<0$

so the graph is concave down