Question

Consider the function f (x) = 2x3 – 5x2 + 3x + 1. (a) find f ′ (x). (3) (b) write down the value of f ′ (2). (1) (c) find the equation of the tangent to the curve of y = f (x) at the point (2, 3).

Answer 1

SOLUTION

GIVEN

Consider the function

$\sf{f(x) = 2 {x}^{3} - 5 {x}^{2} + 3x + 1}$

TO DETERMINE

(a) find f′(x)

(b) write down the value of f ′ (2)

(c) find the equation of the tangent to the curve of y = f (x) at the point (2, 3)

EVALUATION

Here the given function is

$\sf{f(x) = 2 {x}^{3} - 5 {x}^{2} + 3x + 1}$

(i)

$\sf{f(x) = 2 {x}^{3} - 5 {x}^{2} + 3x + 1}$

Differentiating both sides with respect to x we get

$\sf{f'(x) = 6 {x}^{2} - 10x + 3}$

(ii) Putting x = 2 we get

$\sf{f'(2) = 6 \times {2}^{2} - 10 \times 2 + 3}$

$\sf{ \implies \: f'(2) =24- 20 + 3}$

$\sf{ \implies \: f'(2) =7}$

(iii) The required equation of the tangent at the point (2,3) is

$\sf{ \: (y - 3) = f'(2) (x - 2)}$

$\sf{ \implies \: (y - 3) = 7 (x - 2)}$

$\sf{ \implies \: y - 3= 7 x -14}$

$\sf{ \implies \: 7 x -y = 11}$

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