This chapter Application of derivatives mainly features a set of topics just like the rate of change of quantities, Increasing and decreasing functions, Tangents and normals, Approximations, Maxima and minima, and lots more. We are going to discuss the important concepts of the chapter application of derivatives.
1. A function f is said to be
Increasing only if on an interval of (a, b) if x1 < x2
Decreasing on (a,b) if x1 < x2
Let the constant (a, b), if f (x) equals c for all x ∈ (a, b), where c is the constant.
2. First Derivative Test.
3. Second Derivative Test.
The main topics which are covered for the NCERT Solutions for Application of Derivatives Class 12 notes, Chapter 6 are:
Let f be the function which is defined on an interval I. So,
f is claimed to possess a maximum value in I, if there exists some extent c in I such: f(c) > f(x), ∀ x ∈ I. The number f(c) is named the utmost value of f in I and therefore the point c is named to some extent a maximum value of f in I.
f is said to be having a minimum value in I, and if there exists a point c in I so f(c) < f(x), ∀ x ∈ I. The number f(c) is named the minimum value of f in I and therefore the point c is named to some extent the minimum value of f in I.
f is claimed to possess an extreme value in I, if there exists some extent c in I such f(c) is either a maximum value or a minimum value of f in I. The number f(c) is named an extreme value off in I and therefore the point c is named an extreme.
Through the graphs, we will even find the maximum/minimum value of a function to some extent at which it's not even differentiable.
Every monotonic function makes sure that its maximum/minimum value is at the endpoints of the domain of the definition of the function.
Every continuous function on a bounded interval features a maximum and a minimum value.
Let f be a function which is defined on an unbounded interval which is I. Suppose cel is any point. If f has local maxima or local minima at x = c, then either f'(c) = 0 or f isn't differentiable at c.
Critical Point: to some extent c within the domain of a function f at which either f'(c) = 0 or f isn't differentiable, is named a juncture of f.
First Derivative Test: Let f be a function defined on an unbounded interval which is I and f be the continuous of a juncture c in I. So, if f'(x) changes sign from positive to negative as x increases through c, then c may be a point of local maxima.
if f'(x) changes sign from negative to positive as x increases through c, then c may be a point of local minima.
if f'(x) doesn't change sign as x increases through c, then c is neither some extent of local maxima nor some extent of local minima. Such some extent is named some extent of inflection.
Second Derivative Test: Let f(x) be a function that is defined on an interval known as I and c ∈ I. Let f be two times differentiable at c. So,
x = c is a point for the local maxima, if f'(c) equals 0 and f''(c) < 0.
x = c is a point for the local minima, if f'(c) equals 0 and f''(c) > 0.
The test will fail if f'(c) equals 0 and f''(c) equals 0.
Question 1) How do You Calculate Maxima?
Answer) Find the first derivative of a function f(x) and find the critical numbers. Then, find the second derivative of a function f(x) and put the critical numbers. If the value is negative, the function has relative maxima at that point, if the value is positive, the function has relative maxima at that point.
Question 2) What are Derivatives in Calculus?
Answer) Derivatives are a fundamental tool of calculus. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value.
Question 3) What is the Absolute Maximum Value?
Answer) Let f(x) be a function which is defined in its own domain let’s say Z ⊂ R. So, f(x) is claimed to possess the utmost value at some extent a ∈ Z, if f(x) ≤ f(a), ∀ x ∈ Z.
Question 4) Explain Local Maxima and Local Minima.
Ans: (i) A function f(x) is claimed to possess an area maximum value at point x = a, if there exists a neighbourhood (a – δ, a + δ) of a such that f(x) < f(a), ∀ x ∈ (a – δ, a + δ), x ≠ a. Here, f(a) is named the local maximum value of f(x) at the purpose x = a. (ii) A function f(x) is claimed to possess an area minimum value at point x = a, if there exists a neighbourhood (a – δ, a + δ) of a such that f(x) > f(a), ∀ x ∈ (a – δ, a + δ), x ≠ a. Here, f(a) is named the local minimum value of f(x) at x = a.
The points where the function changes its nature from decreasing order to increasing order or vice-versa are called the turning points.