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Chapter 4: Problem 57

Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. $$f(x)=x^{4}-2 x^{3}+1$$

Short Answer

Expert verified
Question: Determine the intervals of concavity and inflection points for the function $$f(x) = x^4 - 2x^3 + 1$$. Answer: The function is concave up in the intervals $(-\infty, 0)$ and $(1, \infty)$, and concave down in the interval $(0, 1)$. The inflection points are at $(0, 1)$ and $(1, 0)$.

Step by step solution

01

Find the first and second derivatives of the function

We have the function $$f(x) = x^4 - 2x^3 + 1$$. We first find the first derivative, $$f'(x)$$. $$ f'(x) = \frac{d}{dx}(x^4 - 2x^3 +1) = 4x^3 - 6x^2 $$ Now, we find the second derivative, $$f''(x)$$. $$ f''(x) = \frac{d^2}{dx^2}(4x^3 - 6x^2) = 12x^2 - 12x $$
02

Find the critical points of the second derivative

To find the critical points of the second derivative, we set $$f''(x)$$ equal to zero. $$ 12x^2 - 12x = 0 $$ Now, we can factor out a common factor of 12. $$ 12(x^2 - x) = 0 $$ Then, we can further factor the expression within the parentheses. $$ 12x(x-1)=0 $$ The critical points are at $$x = 0$$ and $$x = 1$$.
03

Determine the concavity intervals

To determine where the function is concave up or down, we can use the critical points and test the intervals between them in the second derivative. 1. Test $$x = -1$$: $$ f''(-1) = 12(-1)^2-12(-1)= 24>0 $$ 2. Test $$x = 0.5$$: $$ f''(0.5) = 12(0.5)^2 - 12(0.5)= -3<0 $$ 3. Test $$x = 2$$: $$ f''(2) = 12(2)^2 - 12(2)= 24>0 $$ The results show that the function is: 1. Concave up in the interval $$(3,-\infty)$$ 2. Concave down in the interval $$(0,1)$$ 3. Concave up in the interval $$(1,\infty)$$
04

Identify inflection points

We found the critical points at $$x=0$$ and $$x=1$$ in Step 2. These are the points where the graph changes concavity. To find the corresponding y-values, plug the x-values into the original function $$f(x)$$. For x = 0: $$ f(0) = (0)^4 - 2(0)^3 +1 = 1 $$ For x = 1: $$ f(1) = (1)^4 - 2(1)^3+ 1 = 0 $$ So, the inflection points are $$(0,1)$$ and $$(1,0)$$.

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