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

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enhebrevz · Accepted Answer

Step 1It is given thatf′(x)=x2f(x)Substitute x=2f′(2)=22×f(2)Substitute f(2)=10f′(2)=4×10=40f′(x)=x2f(x)Differentiate using the product rulef″(x)=(x2)′×f(x)+x2×f′(x)f″(x)=(2x2−1)×f(x)+x2×f′(x)Substitute x=2f″(2)=2×2×f(2)+22×f′(2)Substitute f(2)=10 and f′(2)=40f″(2)=4×10+4×40=40+160=200

Bertha Jordan · Accepted Answer

Step 1Metod 1:Divide by f(x)f′(x)f(x)=x2Step 2Differentiate both sidesf″(x)f(x)−f′(x)2f(x)2=2xStep 3At x=2f′(2)=22f(2)f′(2)=4×10=40f″(2)f(2)−f′(2)2f(2)2=2×210f″(2)−402102=410f″(2)=400+1600Step 4Solve for f″(x)f″(2)=200010=200Step 5Method 2 (Rose Winter):Use the product rulef″(x)=2xf(x)+x2f′(x)Step 6As before, at x=2 we can calculate:f′(2)=22f(2)f′(2)=4×10=40Step 7Substituting the values we get:f″(2)=2×2×10+22×40=40+160==200

Vasquez · Accepted Answer

Step 1: Let&#039;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&#039;s substitute x=2 into the given information f(2)=10 to find the value of f(2).f(2)=10Finally, 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 · Accepted Answer

Result:f″(2)=200Solution: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 haveddx[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 getf″(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) asf″(2)=2·2·10+22·40Simplifying the expression, we findf″(2)=40+4·40f″(2)=40+160f″(2)=200Therefore, f″(2)=200.

nick1337 · Accepted Answer

Given information:f(2)=10f′(x)=x2f(x)To find f″(2), we can differentiate f′(x) with respect to x and then substitute x = 2. Let&#039;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&#039;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&#039;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)x4We can simplify this equation by multiplying through by x4:x4f′(x)=x2f′(x)−2xf(x)Rearranging the terms:x4f′(x)−x2f′(x)+2xf(x)=0Now, we can substitute x = 2 into the above equation to find f′(2):(2)4f′(2)−(2)2f′(2)+2(2)f(2)=0Simplifying further:16f′(2)−4f′(2)+4f(2)=012f′(2)+4f(2)=0Now, substitute f(2)=10 into the equation:12f′(2)+4(10)=012f′(2)+40=012f′(2)=−40Finally, 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)=40−160f″(2)=−120Therefore, f″(2)=−120.

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

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