Question

In Exercises, find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. $$ f(x)=x^{2}+2 x-4, \quad[-1,1] $$

Short answer

The absolute maximum of \(f(x)\) on the interval \([-1,1]\) is 1 at \(x = -1\), and the absolute minimum is \(-1\) at \(x = 1\).

Step-by-step solution

Calculating the derivative

The derivative of the function \(f(x) = x^2 + 2x - 4\) is calculated as follows: \(f'(x) = 2x + 2\).

Find the critical points

Setting the derivative equals to zero, we can find the critical points. So, we solve for \(x\) in the equation \(2x + 2 = 0\), which yields \(x = -1\).

Evaluate the function

Now we should evaluate the function \(f(x)\) at the critical point and at the endpoints of the interval. So, we have: \(f(-1) = 1\), \(f(1) = -1\), and \(f(-1) = 1\).

Determine the absolute extrema

The largest of the values from Step 3 is the absolute maximum, and the smallest is the absolute minimum. So, we have absolute minimum at \(x = 1\), corresponding to \(f(1) = -1\) and absolute maximum at \(x = -1\), corresponding to \(f(-1) = 1\).

Key concepts

Derivative

The derivative of a function is a fundamental concept in calculus, expressing the rate of change of the function's value with respect to changes in its input. Think of it as the slope of a line tangent to the function at any point. It's a powerful tool for understanding how a function behaves.
The derivative of a polynomial function, like the quadratic function given in our exercise, involves applying basic differentiation rules. Calculating the derivative of a quadratic function such as \(f(x) = x^2 + 2x - 4\) involves using the power rule. The power rule states that if you have a term \(ax^n\), its derivative is \(anx^{n-1}\).
Applying this, we compute the derivative:

• For \(x^2\), its derivative is \(2x\) because \(x^2\) becomes \(2x^{1}\).
• For \(2x\), the derivative is simply \(2\).
• Constants, like \(-4\), become zero in the derivative.

Therefore, the derivative \(f'(x)\) simplifies to \(2x + 2\). This tells us how fast \(f(x)\) is changing with respect to \(x\).

Critical Points

Critical points appear where the derivative is zero or undefined, indicating potential locations of maxima, minima, or points of inflection.
Once you've calculated the derivative, find where this derivative equals zero. This step involves a simple algebraic equation.
In our scenario: \[f'(x) = 2x + 2 = 0\]We solve for \(x\) by isolating the variable.

• Begin by subtracting \(2\) from both sides: \(2x = -2\).
• Then, divide by \(2\) to solve for \(x\), resulting in: \(x = -1\).

So, \(x = -1\) is our critical point. This point is vital because it helps locate potential maximum or minimum values within a given interval.

Interval Evaluation

Interval evaluation involves examining a function's behavior at specific key points: the critical points and the endpoints of a given interval. This step helps us determine absolute extrema, or the highest and lowest points on the graph over a closed interval.
For the function \(f(x) = x^2 + 2x - 4\) on the interval \([-1, 1]\), we evaluated the function at:

• The critical point \(x = -1\)
• The interval's endpoints, \(x = -1\) and \(x = 1\)

Calculating these values involves simple substitution into the original function:

• \(f(-1) = (-1)^2 + 2(-1) - 4 = 1 - 2 - 4 = -5\)
• \(f(1) = (1)^2 + 2(1) - 4 = 1 + 2 - 4 = -1\)

Evaluating helps determine which are the largest and smallest values within the interval. These values correspond to the absolute maximum and minimum values of the function on the interval.

Function Graphing

Graphing a function is an invaluable visual aid that offers a clearer understanding of its behavior, helping to confirm algebraic calculations. For our quadratic function \(f(x) = x^2 + 2x - 4\), graphing can visually indicate the presence of a maximum or minimum at different points within the interval.
With graphing utilities like graphing calculators or software, you can plot the function and easily identify the parabolic shape typical of quadratic functions. Important aspects to observe when graphing include:

• The vertex of the parabola, which often indicates an extremum.
• The general direction in which the parabola opens, upwards or downwards, dictated by the leading coefficients. For \(f(x)\), it opens upwards since \(x^2\) has a positive coefficient.
• The intersection points of the graph with the x-axis, if any, which are the roots of the function.
• The intersections with the endpoints at \([-1, 1]\) and the value at the critical point you calculated.

Using these observations, you can compare them against your calculations to ensure the results match both graphically and algebraically. It provides a holistic view of the function's characteristics over its interval.