Question

Evaluate the expression (x+4)^(2)= and the expression x^(2)+8x+16= for x=-3

Answer 1

The function f(x) has local minimum points at x = a and x = b, and a local maximum point at x = c.

Step-by-step explanation:

The local extrema of a function occur where its derivative is equal to zero or undefined. To find these points, we start by finding the derivative, f'(x), and then setting it equal to zero.

Let's denote the derivative as f'(x). By solving f'(x) = 0, we find the critical points where local extrema may occur.

Next, we evaluate the second derivative, f''(x), at each critical point. If f''(x) > 0, the function has a local minimum at that point. If f''(x) < 0, the function has a local maximum.

Now, let's calculate these values:

1. Find f'(x): This step involves differentiating the function f(x) with respect to x.

[ f'(x) = ... ]

2. Solve for critical points: Set f'(x) = 0 and solve for x.

[ f'(x) = 0 \Rightarrow x = a, b, c ]

3. Evaluate f''(x): Calculate the second derivative.

[ f''(x) = ... ]

4. Determine the nature of extrema: Evaluate f''(x) at each critical point.

- If ( f''(a) > 0 ), there is a local minimum at x = a.

- If ( f''(b) > 0 ), there is a local minimum at x = b.

- If ( f''(c) < 0 ), there is a local maximum at x = c.

In summary, the function f(x) has local minimum points at x = a and x = b, and a local maximum point at x = c, as determined by the second derivative test.

Answer 2

For x = -3, (x+4)^(2) = 1, and x^(2)+8x+16 = 1.

Step-by-step explanation:

Substituting x = -3 into the expressions given, we get:

1. (x+4)^(2) = (-3+4)^(2) = (1)^(2) = 1

2. x^(2)+8x+16 = (-3)^(2) + 8(-3) + 16 = 9 - 24 + 16 = 1

Both expressions, (x+4)^(2) and x^(2)+8x+16, when evaluated for x = -3, yield a value of 1. This result demonstrates that both expressions are equivalent when x is replaced with -3. Substituting -3 into either expression results in the same value of 1.

In mathematics, when solving or evaluating expressions for specific values (in this case, x = -3), it's essential to perform the operations according to the given expression. Substituting the value of -3 into both expressions, we obtain the same result of 1 for both expressions, confirming their equality for this specific value of x.