Question

1st & 2nd derivative test
test to find:
up/down

1. f(x) = \frac{-x^4}{4} + \frac{3x^2}{2} + 2

d) pt of inflections

Explanation:

Step1: Find the first derivative

To find the first derivative of \( f(x) = \frac{-x^4}{4} + \frac{3x^2}{2} + 2 \), we use the power rule. The power rule states that if \( y = x^n \), then \( y' = nx^{n - 1} \).

For the first term \( \frac{-x^4}{4} \), the derivative is \( \frac{-4x^3}{4}=-x^3 \).

For the second term \( \frac{3x^2}{2} \), the derivative is \( \frac{3\times2x}{2}=3x \).

The derivative of the constant term \( 2 \) is \( 0 \).

So, \( f'(x)=-x^3 + 3x \).

Step2: Find the second derivative

Now, we find the second derivative by differentiating \( f'(x) \).

For the term \( -x^3 \), the derivative is \( -3x^2 \).

For the term \( 3x \), the derivative is \( 3 \).

So, \( f''(x)=-3x^2 + 3 \).

Step3: Find the points of inflection

Points of inflection occur where the second derivative changes sign. This happens when \( f''(x) = 0 \) or \( f''(x) \) is undefined. Since \( f''(x) \) is a polynomial, it is defined for all real numbers. So we set \( f''(x)=0 \):

\( -3x^2 + 3 = 0 \)

Divide both sides by \( -3 \):

\( x^2 - 1 = 0 \)

Factor the left - hand side:

\( (x - 1)(x + 1)=0 \)

Setting each factor equal to zero gives \( x - 1 = 0 \) or \( x + 1 = 0 \), so \( x = 1 \) or \( x=-1 \).

Step4: Find the corresponding y - values

Now we find the corresponding \( y \)-values (the values of \( f(x) \)) for \( x = 1 \) and \( x=-1 \).

For \( x = 1 \):

\( f(1)=\frac{-1^4}{4}+\frac{3\times1^2}{2}+2=\frac{-1}{4}+\frac{3}{2}+2 \)

First, find a common denominator, which is \( 4 \).

\( \frac{-1}{4}+\frac{6}{4}+\frac{8}{4}=\frac{-1 + 6+8}{4}=\frac{13}{4}=3.25 \)

For \( x=-1 \):

\( f(-1)=\frac{-(-1)^4}{4}+\frac{3\times(-1)^2}{2}+2=\frac{-1}{4}+\frac{3}{2}+2 \)

Again, using a common denominator of \( 4 \):

\( \frac{-1}{4}+\frac{6}{4}+\frac{8}{4}=\frac{-1 + 6 + 8}{4}=\frac{13}{4}=3.25 \)

Answer:

The points of inflection are \( (-1,\frac{13}{4}) \) and \( (1,\frac{13}{4}) \) or \( (-1, 3.25) \) and \( (1, 3.25) \)