increasing and decreasing functions problems

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Suppose f is an increasing function and g is a decreasing function. Find intervals on which \(f\) is increasing or decreasing.Using the Key Idea 3, we first find the critical values of \(f\). (a) As can be seen from the graph above, [1, 5] is the largest interval on which f is increasing. Lecture 9 - Increasing and Decreasing Functions, Extrema, and the First Derivative Test 9.1 Increasing and Decreasing Functions One of our goals is to be able to solve max/min problems, especially economics related examples. For instance, if \(f\) describes the speed of an object, we might want to know when the speed was increasing or decreasing (i.e., when the object was accelerating vs. decelerating). \(f'\) is never undefined.Since an interval was not specified for us to consider, we consider the entire domain of \(f\) which is \((-\infty,\infty)\). That occurs when \(x=1\), which we've already recognized as an important value.Note: Strictly speaking, \(x=1\) is not a critical value of \(f\) as \(f\) is not defined at \(x=1\). Since we know we want to solve \(f'(x) = 0\), we will do some algebra after taking derivatives.\[\begin{align} f(x) &= x^{\frac{8}{3}}-4x^{\frac{2}{3}} \\ f'(x) &= \dfrac{8}{3} x^{\frac{5}{3}} - \dfrac{8}{3}x^{-\frac{1}{3}} \\ &= \dfrac{8}{3}x^{-\frac{1}{3}} \left(x^{\frac{6}{3}}-1 \right)\\ &=\frac{8}{3}x^{-\frac{1}{3}}(x^2-1)\\ &=\frac{8}{3}x^{-\frac{1}{3}}(x-1)(x+1). (b) Find the largest interval on which f is decreasing. We formalize this concept in a theorem.Let \(f\) be differentiable on \(I\) and let \(c\) be a critical number in \(I\).Find the intervals on which \(f\) is increasing and decreasing, and use the First Derivative Test to determine the relative extrema of \(f\), whereWe start by noting the domain of \(f\): \((-\infty,1)\cup(1,\infty)\). (a) Find the largest interval on which f is increasing. Find the intervals in which a function (given algebraically) is increasing or decreasing. A similar statement can be made for relative minimums. (ii)  What is the largest interval contained in the domain of g on which g is increasing?You can also visit the following web pages on different stuff in math. To find intervals on which \(f\) is increasing and decreasing:We demonstrate using this process in the following example.Let \(f(x) = x^3+x^2-x+1\). Solutions are tractable only through the use of computers to do many calculations for us. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. These three numbers divide the real number line into 4 subintervals:\[(-\infty,-1), \quad (-1, 1), \quad (1,3) \quad \text{and} \quad (3,\infty).\]Pick a number \(p\) from each subinterval and test the sign of \(f'\) at \(p\) to determine whether \(f\) is increasing or decreasing on that interval. Thus h is neither increasing nor decreasing.Here f has domain [0, 4] and g has domain [−1, 5]. The secant line on the graph of \(f\) from \(x=a\) to \(x=b\) is drawn; it has a slope of \((f(b)-f(a))/(b-a)\). At \(x=3\), the sign of f'\ switched from negative to positive, meaning \(f(3)\) is a relative minimum.

Search. Thus f is decreasing. Our previous example demonstrated that this is not always the case. Thus g is increasing. But note:\[\dfrac{f(b)-f(a)}{b-a} \Rightarrow \dfrac{\text{numerator }>0}{\text{denominator } >0} \Rightarrow \text{slope of the secent line} >0 \Rightarrow \text{Average rate of chjange of $f$ on $[a,b]$ is $>0$.} If you're seeing this message, it means we're having trouble loading external resources on our website. Contributions were made by Troy Siemers and Dimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's University. A similar statement can be made for decreasing functions.Our above logic can be summarized as "If \(f\) is increasing, then \(f'\) is probably positive." The other terms are small in comparison, so we know \(f'(-100)>0\). Again, we do well to avoid complicated computations; notice that the denominator of \(f'\) is In summary, \(f\) is increasing on the set \((-\infty,-1)\cup (3,\infty)\) and is decreasing on the set \((-1,1)\cup (1,3)\). All we care about is the sign, so we do not actually have to fully compute \(f'(p)\); pick "nice" values that make this simple.Note we can arrive at the same conclusion without computation. Also, Figure \(\PageIndex{4}\) shows a graph of \(f\), confirming our calculations. (i)  What is the largest interval contained in the domain of f on which f is increasing? (i)  The largest interval on which the function g decreasing is [0, 3].After having gone through the stuff given above, we hope that the students would have understood "Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.Sum and product of the roots of a quadratic equations Sum of the angles in a triangle is 180 degree worksheetDistributive property of multiplication worksheet - IDistributive property of multiplication worksheet - IIDetermine if the relationship is proportional worksheetTrigonometric ratios of angles greater than or equal to 360 degreeDomain and range of inverse  trigonometric functionsWord problems on direct variation and inverse variation Complementary and supplementary angles word problemsWord problems on sum of the angles of a triangle is 180 degreeTranslating the word problems in to algebraic expressionsSum of all three four digit numbers formed with non zero digitsSum of all three four digit numbers formed using 1, 2, 5, 6

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