For what values of x is #f(x)= 2x^3+5x+12# concave or convex?

Answer 1

#f(x)# is concave when #x in ]-oo,0[#
#f(x)# is convex when #x in ]0,+oo[#

We calculate the first and second derivatives

#f(x)=2x^3+5x+12#
#f'(x)=6x^2+5#
#f''(x)=12x#
#f'(x)>0#
#f''(x)=0#, when #x=0#

We draw a chart

#color(white)(aaaa)##Interval##color(white)(aaaaaaa)##]-oo,0[##color(white)(aaaa)##]0,+oo[#
#color(white)(aaaa)##Sign f''(x)##color(white)(aaaaaaaa)##-##color(white)(aaaaaaaa)##+#
#color(white)(aaaa)##function##color(white)(aaaaaaaaaa)##nnn##color(white)(aaaaaaaa)##uuu#

Therefore,

#f(x)# is concave when #x in ]-oo,0[#
#f(x)# is convex when #x in ]0,+oo[#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2
To determine the concavity or convexity of the function \( f(x) = 2x^3 + 5x + 12 \), we need to find its second derivative and then analyze its sign. The second derivative of \( f(x) \) is: \[ f''(x) = 12x \] For concavity, if \( f''(x) > 0 \), the function is concave up (convex) and if \( f''(x) < 0 \), the function is concave down. For convexity, if \( f''(x) > 0 \), the function is convex and if \( f''(x) < 0 \), the function is concave. Since \( f''(x) = 12x \), it's positive for \( x > 0 \) and negative for \( x < 0 \). Therefore: - \( f(x) \) is concave (convex) for \( x > 0 \). - \( f(x) \) is concave down for \( x < 0 \).
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7