If #1/3(x-1)^3+2#, what are the points of inflection, concavity and critical points?

Question

Elijah Abram · Accepted Answer

The point of inflection is #=(1,2)#. The interval of concavity is #x in (-oo,1)#. The interval of convexity is #(1,+oo)#

Let #f(x)=1/3(x-1)^3+2#

Let's develop #f(x)#

#f(x)=1/3(x^3-3x^2+3x-1)+2#

#=x^3/3-x^2+x+5/3#

Calculate the first derivative

#f'(x)=x^2-2x+1=(x-1)^2#

The critical points are when #f'(x)=0#

That is,

#(x-1)^2=0#, #=>#, #x=1#

Now, calculate the second derivative

#f''(x)=2x-2#

The points of inflections are when #f''(x)=0#

That is,

#2x-2=0#, #=>#, #x=1#

Make a sign chart

#color(white)(aaaa)##Interval##color(white)(aaaa)##(-oo,1)##color(white)(aaaa)##(1,+oo)#

#color(white)(aaaa)##Sign f''(x)##color(white)(aaaaa)##-##color(white)(aaaaaaaa)##+#

#color(white)(aaaa)## f(x)##color(white)(aaaaaaaaaaa)##nn##color(white)(aaaaaaaa)##uu#

graph{1/3(x-1)^3+2 [-8.335, 9.445, -2.355, 6.534]}

Aurora Alvarado · Answer

undefined To find the points of inflection, concavity, and critical points of the function $ f(x) = \frac{1}{3}(x-1)^3 + 2 $, follow these steps: 1. **Find the first derivative of the function:** $$ f'(x) = (x-1)^2 $$ 2. **Set the first derivative equal to zero and solve for $ x $ to find critical points:** $$ (x-1)^2 = 0 $$ $$ x - 1 = 0 $$ $$ x = 1 $$ 3. **Find the second derivative of the function:** $$ f''(x) = 2(x-1) $$ 4. **Determine the concavity by analyzing the sign of the second derivative:** - When $ f''(x) > 0 $, the function is concave up. - When $ f''(x) < 0 $, the function is concave down. 5. **Set the second derivative equal to zero to find points of inflection:** $$ 2(x-1) = 0 $$ $$ x - 1 = 0 $$ $$ x = 1 $$ 6. **Now, evaluate the concavity at $ x = 1 $ to determine if it's a point of inflection:** $$ f''(1) = 2(1-1) = 0 $$ - Since $ f''(1) = 0 $, there is no change in concavity at $ x = 1 $, so it's not a point of inflection. 7. **Critical point:** $ x = 1 $ 8. **Concavity:** The function is concave up for $ x < 1 $ and concave down for $ x > 1 $. 9. **Points of Inflection:** There are no points of inflection for the given function.