f ( − x ) f(-x) ? Like seriously?

Calculus Level 3

If 2 f ( x ) + 3 f ( − x ) = x 2 − x + 1 2f(x)+3f(-x)=x^{2}-x+1 , find the value of f ′ ( 2 ) f'(2) .


The answer is 1.8.

Arkajyoti Banerjee by Arkajyoti Banerjee , 20, India

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2 solutions

We have, 2 f ( x ) + 3 f ( − x ) = x 2 − x + 1 2f(x)+3f(-x)=x^{2}-x+1 .

Differentiating both sides with respect to x x , we have:

2 f ′ ( x ) − 3 f ′ ( − x ) = 2 x − 1 2f'(x)-3f'(-x)=2x-1

Now, when x = 2 x=2 ,

2 f ′ ( x ) − 3 f ′ ( − x ) = 2 x − 1 ⟹ 2 f ′ ( 2 ) − 3 f ′ ( − 2 ) = 2 ( 2 ) − 1 = 3 ⟹ 2 f ′ ( 2 ) − 3 f ′ ( − 2 ) = 3 2f'(x)-3f'(-x)=2x-1 \\ \implies 2f'(2)-3f'(-2)=2(2)-1=3 \\ \implies \boxed{2f'(2)-3f'(-2)=3}

Similarly, when x = − 2 x=-2 ,

2 f ′ ( x ) − 3 f ′ ( − x ) = 2 x − 1 ⟹ 2 f ′ ( − 2 ) − 3 f ′ ( 2 ) = 2 ( − 2 ) − 1 = − 5 ⟹ 3 f ′ ( 2 ) − 2 f ′ ( 2 ) = 5 2f'(x)-3f'(-x)=2x-1 \\ \implies 2f'(-2)-3f'(2)=2(-2)-1=-5\\ \implies \boxed{3f'(2)-2f'(2)=5}

Solving the above boxed equations for f ′ ( 2 ) f'(2) , we have f ′ ( 2 ) = 9 5 = 1.8 f'(2)=\frac{9}{5}=\boxed{1.8}

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It will be better to find f ( x ) \text{It will be better to find } f(x)

2 f ( x ) + 3 f ( − x ) = x 2 − x + 1 ⟹ 4 f ( x ) + 6 f ( − x ) = 2 x 2 − 2 x + 2 x → − x 2 f ( − x ) + 3 f ( x ) = x 2 + x + 1 ⟹ − 6 f ( − x ) − 9 f ( x ) = − 3 x 2 − 3 x − 3 \quad 2f(x)+3f(-x) = x^2-x+1 \implies 4f(x)+6f(-x)=2x^2-2x+2 \\ \quad x \rightarrow -x \\ \quad 2f(-x)+3f(x) = x^2+x+1 \implies -6f(-x)-9f(x)=-3x^2-3x-3

Add these two equations 5 f ( x ) = x 2 + 5 x + 1 f ′ ( x ) = 1 5 ( 2 x + 5 ) \quad \text{Add these two equations } \\ \quad 5f(x)= x^2+5x+1 \\ \quad \boxed{f^{'}(x) = \dfrac15 \left( 2x+5 \right)}

Sabhrant Sachan - 4 years, 11 months ago

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Since it is asked to find out the value of f ′ ( 2 ) f'(2) , I directly went for finding f ′ ( 2 ) f'(2) .

Arkajyoti Banerjee - 4 years, 11 months ago
Tamir Dror
Jul 7, 2016

Easier solution is to the following: 1. note f(x) must be a polynom with degree 2 2. Write the polynom as Ax^2+Bx (the constant is not needed since the diffrentiation is 0) 3. Find A and B 4. Diffrentiate 5. Substitute x=2 6. Easy ha? :p

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