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How to find relative extrema using the First Derivative Test?
- Find critical points. 2. Determine the sign of the derivative on either side of each critical point. 3. Apply the First Derivative Test rules.
Steps to find critical points of a function.
- Find the derivative of the function. 2. Set the derivative equal to zero and solve for x. 3. Find where the derivative is undefined.
How to determine if a function is increasing on an interval?
- Find the derivative. 2. Determine where the derivative is positive.
How to determine if a function is decreasing on an interval?
- Find the derivative. 2. Determine where the derivative is negative.
Steps to find the relative minima of (h(x)) given (h'(x) = x^2(x-3)(x+4)).
- Set (h'(x) = 0) to find critical points: (x = -4, 0, 3). 2. Test intervals around each critical point to determine the sign change of (h'(x)). 3. Identify where (h'(x)) changes from negative to positive.
Steps to find the relative maxima of (g(x) = x^5 - 80x).
- Find (g'(x) = 5x^4 - 80). 2. Set (g'(x) = 0) to find critical points: (x = -2, 2). 3. Test intervals around each critical point to determine the sign change of (g'(x)). 4. Identify where (g'(x)) changes from positive to negative.
How do you test the intervals around a critical point?
Choose a test value within each interval and evaluate the derivative at that value. The sign of the derivative indicates whether the function is increasing or decreasing in that interval.
What should you do if a critical point is an endpoint of the domain?
Evaluate the derivative only on the side of the critical point that is within the domain. The sign of the derivative on that side determines if the endpoint is a local extremum.
How do you find the y-coordinate of a local extremum?
Substitute the x-coordinate of the local extremum into the original function (f(x)) to find the corresponding y-coordinate.
How do you handle critical points where the derivative is undefined?
Treat them similarly to where the derivative is zero: test the intervals around the critical point to determine the sign change of the derivative.
Define relative (local) maximum.
A point where the function's derivative changes from positive to negative.
Define relative (local) minimum.
A point where the function's derivative changes from negative to positive.
What is a critical point?
A point where the derivative of the function is equal to 0 or undefined.
Define the First Derivative Test.
A method to determine relative extrema by analyzing the sign change of the first derivative around critical points.
What is an extremum?
A maximum or minimum value of a function.
What is the significance of a critical point?
Critical points are potential locations for relative maxima or minima.
Define increasing function in terms of its derivative.
A function is increasing where its derivative is positive.
Define decreasing function in terms of its derivative.
A function is decreasing where its derivative is negative.
What does it mean if the derivative is zero at a point?
It indicates a horizontal tangent, which could be a local extremum or a saddle point.
What is a polynomial function?
A function that can be expressed in the form (f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0), where n is a non-negative integer and the (a_i) are constants.
What does a horizontal tangent line on the graph of (f(x)) indicate?
A critical point where (f'(x) = 0), which could be a local maximum, local minimum, or saddle point.
How can you identify local extrema on the graph of (f(x))?
Look for points where the graph changes direction from increasing to decreasing (local max) or decreasing to increasing (local min).
How does the graph of (f'(x)) relate to the increasing/decreasing behavior of (f(x))?
Where (f'(x) > 0), (f(x)) is increasing. Where (f'(x) < 0), (f(x)) is decreasing.
What does the x-intercept of the graph of (f'(x)) represent?
It represents a critical point of the original function (f(x)).
How can you determine if a critical point is a local maximum from the graph of (f'(x))?
If (f'(x)) changes from positive to negative at the x-intercept, the corresponding point on (f(x)) is a local maximum.
How can you determine if a critical point is a local minimum from the graph of (f'(x))?
If (f'(x)) changes from negative to positive at the x-intercept, the corresponding point on (f(x)) is a local minimum.
What does it mean if (f'(x)) is always positive?
(f(x)) is always increasing.
What does it mean if (f'(x)) is always negative?
(f(x)) is always decreasing.
How can you identify intervals where (f(x)) is increasing or decreasing by looking at the graph of (f'(x))?
If (f'(x)) is above the x-axis, (f(x)) is increasing. If (f'(x)) is below the x-axis, (f(x)) is decreasing.
How does the steepness of the graph of (f(x)) relate to the value of (f'(x))?
The steeper the graph of (f(x)), the larger the absolute value of (f'(x)).