# What is the equation of the line tangent to # f(x)=2x^2 + cos(x)# at # x=pi/3#?

Calculate the derivative.

In point-slope form, write:

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The equation of the line tangent to f(x)=2x^2 + cos(x) at x=pi/3 is y = 3√3x - 2√3π + 2/3.

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To find the equation of the line tangent to the function ( f(x) = 2x^2 + \cos(x) ) at ( x = \frac{\pi}{3} ), we need to follow these steps:

- Find the derivative of the function ( f(x) ) using differentiation rules.
- Evaluate the derivative at ( x = \frac{\pi}{3} ) to find the slope of the tangent line.
- Use the point-slope form of the equation of a line, where the slope is the derivative evaluated at ( x = \frac{\pi}{3} ), and the point is ( (x, f(x)) ) where ( x = \frac{\pi}{3} ).

Step 1: Find the derivative of ( f(x) ): [ f'(x) = 4x - \sin(x) ]

Step 2: Evaluate the derivative at ( x = \frac{\pi}{3} ): [ f'\left(\frac{\pi}{3}\right) = 4\left(\frac{\pi}{3}\right) - \sin\left(\frac{\pi}{3}\right) ] [ = \frac{4\pi}{3} - \frac{\sqrt{3}}{2} ]

Step 3: Use the point-slope form to find the equation of the tangent line: [ y - f\left(\frac{\pi}{3}\right) = f'\left(\frac{\pi}{3}\right)(x - \frac{\pi}{3}) ] [ y - \left(2\left(\frac{\pi}{3}\right)^2 + \cos\left(\frac{\pi}{3}\right)\right) = \left(\frac{4\pi}{3} - \frac{\sqrt{3}}{2}\right)\left(x - \frac{\pi}{3}\right) ]

This is the equation of the tangent line to ( f(x) ) at ( x = \frac{\pi}{3} ).

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