Question

{Use of Tech} Critical points and extreme values

a. Find the critical points of the following functions on the given interval. Use a root finder, if necessary.
b. Use a graphing utility to determine whether the critical points correspond to local maxima, local minima, or neither.
c. Find the absolute maximum and minimum values on the given interval, if they exist

h(x) (5-x)/(x² + 2x - 3) on [-10,10]

Accepted Answer

To find the critical points of the function $$ h(x) = \frac{5-x}{x^2 + 2x - 3} $$, first compute the derivative $$ h'(x) . $$Use the quotient rule: $$ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} $$, where $$ u = 5-x $$ and $$ v = x^2 + 2x - 3 . $$Calculate the derivatives: $$ u' = -1 $$ and $$ v' = 2x + 2 . $$Substitute these into the quotient rule to find $$ h'(x) = \frac{(-1)(x^2 + 2x - 3) - (5-x)(2x + 2)}{(x^2 + 2x - 3)^2} . $$Simplify the expression to find $$ h'(x) . $$Set $$ h'(x) = 0  to $$find the critical points. This involves solving the equation $$ (-1)(x^2 + 2x - 3) - (5-x)(2x + 2) = 0 . $$Use algebraic manipulation or a root finder to solve for $$ x . $$Use a graphing utility to plot $$ h(x) $$ and visually inspect the critical points. Determine whether each critical point is a local maximum, local minimum, or neither by observing the behavior of the graph around these points. To find the absolute maximum and minimum values on the interval $$[-10, 10]$$, evaluate $$ h(x)  at $$the critical points and at the endpoints $$ x = -10 $$ and $$ x = 10 . $$Compare these values to determine the absolute extrema.

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Concetti chiave

Critical Points

Local Extrema

Absolute Extrema