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

Carla Murphy

Carla Murphy

Answered

2021-12-21

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

Answer & Explanation

enhebrevz

enhebrevz

Expert

2021-12-22Added 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

Expert

2021-12-23Added 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

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