(1 point) ldentity he graphs A (blue, B(red) and C (green) as the graphs of a fu

Question

(1 point) ldentity he graphs A (blue, B(red) and C (green) as the graphs of a function f(z) and ts dernvatves (a) and () cicking on the sketch will give you a version of the picture in a separate window.) is the graph of the function, f(r) s the graph of the functions first derivative, f'(z) is the graph of the function's second derivative, f" (r) Hint: Remember that f(z) s tseff a function, and we can find the derivative of the function f(x), whch is called the second derivative of the function f(x) and denoted by f"()

Explanation / Answer

# by using 1st derivative

to identify graph of a function f(x) and graph of its derivative f'(x) we can bank on following relation

1) f'x(x) > 0 => f(x) is an increasing function i.e. if graph of (x) going upward then graph of f'(x) will remain above x - axis

2)f'x(x) < 0 => f(x) is an decreasing function i.e. if graph of (x) going downward then graph of f'(x) will remain below x - axis

3) f'(x) = 0 => f(x) will attain maximam or minimam point i.e. when f(x) in it maxima or minima the f'(x) will cross x-axis

if we compare between the graph give in the question

a) between green and blue .

1) green curve is above x - axis when blue curve is going upward

2) green curve is below x axis when blue curve is going downward

3) green curve crossing x-axis when blue curve is in its maxima /minima

so from above observation we can conclude green curve is graph of differentiation of the function represent by blue curve

b) now compare between green and red curve

1) green curve is above x - axis when blue curve is going upward

2) red curve is below x axis when green curve is going downward

3) red curve crossing x-axis when green curve is in its maxima /minima

so from above observation we can conclude red curve is graph of differentiation of the function represent by green curve

# by using second derivative

1) f'' (x)> 0 => f(x) concave upward ( whole curve will lie above tangent drawn at any point in that interval )

2) f''(x) < 0 => f(x) concave downward ( whole curve will lie below tangent drawn at any point in that interval )

3) f''(x) = 0 => may be point of inflection for f(x) ( where f(x) move from concave upward to concave down ward or vice versa )

above relation is exist between blue and red curve

so finally we can conclude blue carve represents original function [ f(x) ] , green curve represents its derivative [ f'(x) ] , and red curve reprezents its double derivative [ f''(x) ]

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