For what values of x is #f(x)=4x^5-5x^4# concave or convex?

Answer 1

The answer is #f(x)# is concave down for #x in ]-oo, 1]# and concave up when # x in [1, +oo[#

We calculate the first derivative and we build a sign chart

#f(x)=4x^5-5x^4#
#f'(x)=20x^4-20x^3#
#f'(x)=20x^3(x-1)#
The critical points are when #f'(x)=0#
#20x^3(x-1)=0#
#x=0# and #x=1#

Now we construct the sign chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaaaaa)##0##color(white)(aaaaaa)##1##color(white)(aaaaa)##+oo#
#color(white)(aaaa)##x##color(white)(aaaaaaaa)##-##color(white)(aa)####color(white)(aaaa)##+##color(white)(aaaa)##+#
#color(white)(aaaa)##x-1##color(white)(aaaaa)##-##color(white)(aaaaaa)##-##color(white)(aaaa)##+#
#color(white)(aaaa)##f'(x)##color(white)(aaaaa)##+##color(white)(aaaaaa)##-##color(white)(aaaa)##+#
#color(white)(aaaa)##f(x)##color(white)(aaaaaa)##↗^(0)##color(white)(aaaa)##↘_(-1)##color(white)(aaaa)##↗^(+oo)#

Therefore,

#f(x)# is concave down for #x in ]-oo, 1]# and concave up when
# x in [1, +oo[#
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Answer 2

To determine the concavity of ( f(x) = 4x^5 - 5x^4 ), you need to find the second derivative of ( f(x) ) and then examine its sign. If the second derivative is positive, the function is concave up (convex), and if it's negative, the function is concave down. The second derivative of ( f(x) ) is ( f''(x) = 80x^3 - 60x^2 ). To find the values of ( x ) for which the function is concave or convex, set ( f''(x) ) equal to zero and solve for ( x ). Then, test the intervals between the critical points to determine the sign of ( f''(x) ).

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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.

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