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Description
When you're looking for extrema in a closed interval, you're looking for the highest and lowest points that the function attains in the interval, including at the endpoints of the interval. For that reason, the candidates for extrema will always be only any critical points that lie inside the interval and the endpoints of the interval.
Description
To arrive at the Mean Value Theorem, we first need to understand Rolle's Theorem. Rolle's Theorem applies to functions on a closed interval, where the function is differentiable on the open interval, and where the value of the function at the endpoints of the interval are equal to one another. For functions of this type, Rolle's Theorem tells us that there is a point inside the interval where the slope of the function's derivative is equal to 0; in...
8) Two Real Numbers with Minimum Product: Calculus-Applications of Derivatives: Applied Optimization
Description
Minimize the product of two real numbers while their difference is constrained.
Description
In the same way that critical points indicate where a function changes direction, inflection points indicate where a function changes concavity. If a function is concave down (curving downwards like a rainbow) and hits an inflection point, it'll become concave up (curving upwards like a bowl). Conversely, if a function is concave up and hits an inflection point, it'll become concave down.
Description
L'Hospital's rule is used to get you out of sticky situations with indeterminate limit forms. If you plug into the function the number you're approaching and your result is indeterminate, you should apply L'Hospital's rule. To use it, take the derivatives of the numerator and denominator and replace the original numerator and denominator with their derivatives. Then plug in the number you're approaching. If you still get an indeterminate form, continue...
Description
The critical points of a function are the points where the function changes direction. If the function was increasing and reaches a critical point, it starts decreasing there. Conversely, if a function was decreasing and reaches a critical point, then it starts increasing there. Therefore, the critical points of a function are the points that represent local maxima and minima of the function (its extrema). To find critical points, Take the derivative...
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