Increasing and Decreasing Functions-<\/strong> We shall first learn about increasing functions<\/p>\n Increasing Function-<\/strong><\/p>\n (a) Strictly increasing function- <\/strong>A function <\/span>f <\/em>(x) is said to be a strictly increasing function on <\/span>(a, b) <\/em>if x<\/span>1<\/sub>< x2\u00a0<\/span> Classification of Strictly Increasing Functions on the basis of Shape-<\/strong><\/p>\n (i) Concave up<\/strong>– When f'(x)>0 and f”(x)>0 for all x\u00a0 Refer to the graph in below-given figure\u00a0 (ii) When f'(x)>0 and f”(x)=0 for all x\u00a0 Refer to the graph in below-given figure\u00a0 (iii) Concave down or convex up –<\/strong> When f'(x)>0 and f”(x)<0 for all x\u00a0 Refer to the graph in below-given figure\u00a0 (b) Only Increasing or non-decreasing functions- <\/strong>A function is said to be non-decreasing if for \u00a0 and for the portion BC ; \u00a0 thus overall we can say that \u00a0 Refer to the graph in below-given figure\u00a0 Decreasing Functions-<\/strong><\/p>\n (a) Strictly Decreasing Function-<\/strong>A function f <\/em>(x) is said to be a strictly increasing function \u00a0on\u00a0(a, b) <\/em>if x1<\/sub>< x2 \u00a0 Thus, f (x) <\/em>is strictly decreasing on (a, b) <\/em>if the values of f\u00a0(x) \u00a0de<\/em>crease with the increase in the values of x.<\/p>\n <\/p>\n Graphically, f (x) <\/em>is a decreasing function on\u00a0\u00a0(a, b) <\/em>if the graph y = f (x) <\/em>moves down as x moves to the right. The graph in Fig. is the graph of a strictly decreasing function on \u00a0 \u00a0 \u00a0(a, b). Refer to the graph in below-given figure\u00a0 Classification of Strictly Decreasing Functions on the basis of Shape-<\/strong><\/p>\n (i) Concave up or Convex down-<\/strong>When f'(x)<0 and f”(x)>0 for all x\u00a0 Refer to the graph in below-given figure\u00a0 (ii) When f'(x)<0 and f”(x)=0 for all x\u00a0 Refer to the graph in below-given figure\u00a0 (iii) Concave down or convex up –<\/strong> When f'(x)<0 and f”(x)<0 for all x\u00a0 Refer to the graph in below-given figure\u00a0 (b) Only decreasing or non-increasing functions- <\/strong>A function is said to be non-increasing if for \u00a0 and for the portion BC ; \u00a0 thus overall we can say that \u00a0 Obviously for this \u00a0 Refer to the graph in below-given figure\u00a0 Monotonic Function: <\/strong>A function is said to be monotonic in the given interval (a,b) If it is either increasing or decreasing in the given interval.<\/p>\n Definition: <\/strong>A function is said to be increasing (decreasing) at a point x0 if there is an \u00a0 \u00a0 \u00a0 interval (x0<\/sub>-h,x0<\/sub>+h)\u00a0containing x0 such that f(x) is increasing(decreasing) on (x<\/span>0<\/sub>-h,x<\/span>0<\/sub>+h)<\/span><\/p>\n Definition: <\/strong>A function is said to be increasing or decreasing on [a,b] if it is increasing(decreasing) at (a,b) and it is also increasing (decreasing) at x=a and x=b<\/p>\n Conditions for Increasing and Decreasing Functions- <\/strong>We can easily identify \u00a0increasing and decreasing functions with the help of differentiation.<\/p>\n (i) If f'(x)>0 for all x (ii) If f'(x)<0 for all x Properties of Monotonic Functions-<\/strong><\/p>\n (a) If f(x) is a function that is strictly increasing in the interval [a,b] then inverse of given function (f-1<\/sup>) exists and \u00a0f-1 \u00a0<\/sup>is also strictly increasing function<\/p>\n (b)\u00a0If f(x) and g(x) are functions that are strictly increasing or decreasing in the interval [a,b] then composite of given functions gof(x) is \u00a0also strictly increasing or decreasing<\/p>\n (c) If f(x) increasing function then its reciprocal 1\/f(x) is decreasing function<\/p>\n (d) If f(x) and g(x) are both increasing functions then (f+g)(x) is also increasing<\/p>\n (e) If f(x) is increasing and g(x) is decreasing or vice-versa then gof(x) is decreasing<\/p>\n <\/p> f(x1<\/sub>) < f <\/em>(x<\/span>2<\/sub>) for all x<\/span>l<\/sub>, x<\/span>2<\/sub>
(a, b).<\/em>Thus, f(x) <\/em>is strictly increasing on (a, b) <\/em>if the values of f(x) <\/em>increase with the increase in the values of x.Refer to the graph in below-given figure
Graphically, f (x) <\/em>is increasing on (a, b) <\/em>if the graph y = f (x) <\/em>moves up as x moves to the right. The graph in Fig.1 \u00a0is the graph of a strictly increasing function on (a, b).\u00a0<\/em><\/p>\n
\u00a0domain<\/p>\n
<\/p>\n
\u00a0domain<\/p>\n
<\/p>\n
\u00a0domain<\/p>\n
<\/p>\n
\u00a0
\u00a0
\u00a0as shown in the graph, for AB and CD;<\/p>\n
\u00a0 \u00a0\u00a0
\u00a0\u00a0<\/span>
<\/p>\n
\u00a0 \u00a0 \u00a0
<\/p>\n
\u00a0 \u00a0
\u00a0so it is obvious for increasing or non-decreasing functions, f'(x)\u00a0
0 with equality holding in interval like BC.<\/p>\n
<\/p>\n
f (x1<\/sub>) > f <\/em>(x2<\/sub>) for all xl<\/sub>, x2<\/sub>
(a, b)<\/em><\/p>\n
<\/em><\/p>\n
\u00a0domain<\/p>\n
<\/em><\/p>\n
\u00a0domain<\/p>\n
<\/em><\/p>\n
\u00a0domain<\/p>\n
<\/em><\/p>\n
\u00a0 \u00a0\u00a0
\u00a0
\u00a0as shown in the graph, for AB and CD ;<\/p>\n
\u00a0 \u00a0\u00a0
\u00a0\u00a0
<\/p>\n
\u00a0 \u00a0 \u00a0
<\/p>\n
\u00a0\u00a0 \u00a0
<\/p>\n
\u00a0where equality holds for horizontal part of the graph i.e, in the interval of BC.<\/span><\/p>\n
<\/em><\/p>\n
\u00a0(a,b), then f(x) is increasing on (a,b)<\/p>\n
\u00a0(a,b), then f(x) is decreasing on (a,b)<\/p>\n
Here I am posting a worksheet on Increasing and Decreasing Functions and another worksheet with 200 (approx.) questions on Applications of Derivatives.<\/h3>\n
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![\"ib](\"http:\/\/ibelitetutor.com\/blog\/wp-content\/uploads\/2018\/04\/ib-free-demo-class.png\")
<\/p>\nWhatsapp at +919911262206 or fill the form<\/h5>\n\n
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