If f(2)=10 and f'(x)=x^{2}f(x) for all x, find f''(2)

Carla Murphy

Carla Murphy

Answered question

2021-12-21

If f(2)=10 and f(x)=x2f(x) for all x, find f(2)

Answer & Explanation

enhebrevz

enhebrevz

Beginner2021-12-22Added 25 answers

Step 1
It is given that
f(x)=x2f(x)
Substitute x=2
f(2)=22×f(2)
Substitute f(2)=10
f(2)=4×10=40
f(x)=x2f(x)
Differentiate using the product rule
f(x)=(x2)×f(x)+x2×f(x)
f(x)=(2x21)×f(x)+x2×f(x)
Substitute x=2
f(2)=2×2×f(2)+22×f(2)
Substitute f(2)=10 and f(2)=40
f(2)=4×10+4×40=40+160=200
Bertha Jordan

Bertha Jordan

Beginner2021-12-23Added 37 answers

Step 1
Metod 1:
Divide by f(x)
f(x)f(x)=x2
Step 2
Differentiate both sides
f(x)f(x)f(x)2f(x)2=2x
Step 3
At x=2
f(2)=22f(2)
f(2)=4×10=40
f(2)f(2)f(2)2f(2)2=2×2
10f(2)402102=4
10f(2)=400+1600
Step 4
Solve for f(x)
f(2)=200010=200
Step 5
Method 2 (Rose Winter):
Use the product rule
f(x)=2xf(x)+x2f(x)
Step 6
As before, at x=2 we can calculate:
f(2)=22f(2)
f(2)=4×10=40
Step 7
Substituting the values we get:
f(2)=2×2×10+22×40=
40+160=
=200
Vasquez

Vasquez

Expert2023-06-17Added 669 answers

Step 1: Let's begin by finding the first derivative of the function f(x). We are given that f(x)=x2f(x).
f(x)=x2f(x)
Next, we need to find the second derivative f(x). To do this, we differentiate the expression f(x) with respect to x.
f(x)=ddx(x2f(x))
Step 2: Now, let's substitute x=2 into the given information f(2)=10 to find the value of f(2).
f(2)=10
Finally, we can evaluate f(2) by substituting x=2 into the expression we obtained for f(x).
f(2)=ddx(x2f(x))|x=2
Don Sumner

Don Sumner

Skilled2023-06-17Added 184 answers

Result:
f(2)=200
Solution:
We are given that f(2)=10 and f(x)=x2f(x) for all x. We need to find f(2).
To find f(2), we can differentiate the given expression f(x)=x2f(x) with respect to x.
Differentiating both sides of the equation, we have
ddx[f(x)]=ddx[x2f(x)]
Using the chain rule, the left side becomes f(x) and the right side becomes 2xf(x)+x2f(x).
Therefore, we have f(x)=2xf(x)+x2f(x).
Now, substituting x=2 into the equation, we get
f(2)=2·2·f(2)+22·f(2)
Substituting the given values f(2)=10 and f(2)=22·f(2)=4·10=40, we can calculate f(2) as
f(2)=2·2·10+22·40
Simplifying the expression, we find
f(2)=40+4·40
f(2)=40+160
f(2)=200
Therefore, f(2)=200.
nick1337

nick1337

Expert2023-06-17Added 777 answers

Given information:
f(2)=10
f(x)=x2f(x)
To find f(2), we can differentiate f(x) with respect to x and then substitute x = 2. Let's start by finding f(x) using the chain rule:
ddx[f(x)]=ddx[x2f(x)]
To apply the chain rule, we differentiate f(x) with respect to x while treating x2 as a constant:
f(x)=2xf(x)+x2f(x)
Now, let's substitute x = 2 into the above equation to find f(2):
f(2)=2(2)f(2)+(2)2f(2)
Using the given information that f(2)=10, we can substitute the values into the equation:
f(2)=2(2)(10)+(2)2f(2)
Simplifying further:
f(2)=40+4f(2)
We still need the value of f(2). Let's differentiate f(x) using the given information that f(x)=x2f(x):
ddx[f(x)]=ddx[f(x)x2]
Applying the quotient rule:
f(x)=x2f(x)2xf(x)x4
We can simplify this equation by multiplying through by x4:
x4f(x)=x2f(x)2xf(x)
Rearranging the terms:
x4f(x)x2f(x)+2xf(x)=0
Now, we can substitute x = 2 into the above equation to find f(2):
(2)4f(2)(2)2f(2)+2(2)f(2)=0
Simplifying further:
16f(2)4f(2)+4f(2)=0
12f(2)+4f(2)=0
Now, substitute f(2)=10 into the equation:
12f(2)+4(10)=0
12f(2)+40=0
12f(2)=40
Finally, we can substitute the value of f(2) back into the equation for f(2):
f(2)=40+4f(2)
f(2)=40+4(40)
f(2)=40160
f(2)=120
Therefore, f(2)=120.

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