CalculusFunction
posted by Liliana .
sketch the graph of a function in neighborhood x=2 that satisfies these conditions
f(2)=3
f'(2)=2
f"(2)=1
you can find a specific function that satisfies all of the given conditions
i know the first derivative gives you critical points, and second derivative gives you points of inflection/concavity but how would you do this. i have no clue where to start. thankyou

CalculusFunction 
Steve
There are lots of possibilities here. The easiest is probably a simple parabola, with constant f" = 1
f' = x + c
since f'(2) = 2,
2 = 2 + c
c=4
f' = x+4
f = 1/2 x^2 + 4x + c
since f(2) = 3,
3 = 2 + 8 + c
c = 3
f(x) = 1/2 x^2 + 4x  3
Or, you could try an exponental
f" = e^x+c
1 = e^2+c
c = 1e^2
f" = e^x 1e^2
f' = e^x  (1+e^2)x + c
2 = e^2  (1+e^2) + c
c = 3
f' = e^x  (1+e^2)x + 3
f = e^x  1/2 (1+e^2)x^2 + 3x + c
3 = e^2  2(1+e^2) + 6 + c
c = e^21
f(x) = e^x  1/2 (1+e^2)x^2 + 3x + e^21
You could do the same for trig functions, too, working with a translated graph:
f(x) = asin(b(x2)+c)
f' = abcos(b(x2)+c)
f" = ab^2 cos(b(x2)+c)
3 = asin(c)
2 = abcos(c)
1 = ab^2 cos(c)
b = 1/2
2/3 = 1/2 cot(c)
so, cot(c) = 4/3
so, 3 = a (4/5)
c = 15/4
f(x) = 15/4 sin(1/2(x2)arccot(4/3))
Not sure on this one, but you see the method.
Respond to this Question
Similar Questions

graphing exponential function
P(x) = (xa)^4 (xb)^3 (xc) where a>0, b<0, c>0, a<c How do I sketch a graph of this? 
Calculus
Sketch a graph of a function f(x) that is differentiable and that satisfies the following conditions. c) f'(3) = 0 and f'(1) = 0 Please show me, step by step, how to sketch the problem! 
Math
Sketch the graph of a function f that satisfies all the following conditions. (a) f(0)=0,f'=1 (b) f''(x)>0 for x<0, f''<0 for x>0 
calculus
sketch graph of a function f that is differentiable and that satisfies the following 
calculus
sketch graph of a function f that is differentiable and that satisfies the following conditions:(1)f'(x)>0, when x<5 (2)f'(x)<0, when 5<x<1 and when x>1 (3)f'(5) =0and f'(1)=0 (4)f(5)=6and f(1)=2. 
math
Write a linear function that satisfies the given conditions: f(0)=12, F(3)= 2 
Algebra
I could use some help on this please. Write a linear function that satisfies the given conditions: h(1) = 4, h(0) = 5 Thanks!!! 
Math
The twice–differentiable function f is defined for all real numbers and satisfies the following conditions: f(0)=3 f′(0)=5 f″(0)=7 a)The function g is given by g(x)=e^ax+f(x) for all real numbers, where a is a constant. … 
Algebra
Find a rational function that satisfies the given conditions. Vertical asymptotes x=2,x=7 Horizontal asymptote y=7/2 xintercept (−5, 0) 
Precalc
Find an equation of a rational function that satisfies the given conditions. Vertical asymptote: x = 4 Horizontal asymptote: y = 0 Xintercepts: none Yintercepts (0, 1/4) Multiplicity of 1 Removable discontinuity: x = 1 End behavior: …