For what values of x is #f(x)=4x^3-3x+5# concave or convex?
1 Answer
Jan 17, 2016
Explanation:
Convexity and concavity are determined by the sign of the second derivative.
- If
#f''(a)>0# , then#f(x)# is convex at#x=a# . - If
#f''(a)<0# , then#f(x)# is concave at#x=a# .
Find the second derivative of the function.
#f(x)=4x^3-3x+5#
#f'(x)=12x^2-3#
#f''(x)=24x#
Analyze the sign of the second derivative,
#f''(x)<0# when#x<0# .#f''(x)>0# when#x>0# .
Thus,
#f(x)# is convex on#(0,+oo)# .#f(x)# is concave on#(-oo,0)# .
