Question

is the graph y=sin(x^4) increasing or decreasing at x=10 and is it concave up or concave down

Answer 1

y = sin (x^4)
y' = 4 x^3 cos (x^4)
y'(10) = 4000 cos (10000)
cos 10000 = 0.17..... > 0
==> y' > 0
==> increasing

Answer 2

what about the convaveness

Answer 3

Find y''. If y'' (10) > 0, then it is concave up and vice versa.
In this case, using product rule,
y'' = 4 [(x^3)(4x^3)(-sin (x^4)) + (cos (x^4))(12x^2)]
y'' = 4[12x^2 cos(x^4) - 4x^6 sin(x^4)]
y'' = 48x^2 cos(x^4) - 16x^6 sin(x^4)
y''(10) = 4800cos(10000) - 16000000sin(10000)
y''(10) = 4 885 259.88... > 0
Since y''(10) > 0, concave up at x = 10

Answer 4

THANKS!!!