Maxi.......Mom

Calculus Level 3

f ( x ) = x 3 3 − 3 2 x 2 + 2 x + 5 \large f(x) = \frac{x^3}{3} - \frac32x^2 + 2x + 5

Find the local maximum value M M of the function above. Give your answer as 6 M 6M .


The answer is 35.

Nikhil Raj by Nikhil Raj , 19, India

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1 solution

Nikhil Raj
May 31, 2017

Here, f ( x ) = x 3 3 − 3 2 x 2 + 2 x + 5 S o , f ′ ( x ) = x 2 − 3 x + 2 For finding maximum and minimum value, we need to find the points whose value is at extrema. For finding points, f ′ ( x ) = 0 ⟹ x 2 − 3 x + 2 = 0 ⟹ ( x − 2 ) ( x − 1 ) = 0 ⟹ x = 1 , 2 Now we will identify which point has maximum value, for that we will first find f ′ ′ ( x ) . S o , f ′ ′ ( x ) = 2 x − 3 Putting x = 1 , 2 in f ′ ′ ( x ) , we have f ′ ′ ( 1 ) = − 1 < 0....................... ( Since, value is negative so given function has maximum value at x = 1 ) N o w , f ′ ′ ( 2 ) = 1 > 0................... ( Since, value is positive so given function has minimum value at x = 2 ) We need to find the maximum value of f ( x ) , S o , Max. value = f ( 1 ) = 1 3 − 3 2 + 2 + 5 = 35 6 . ∴ A n s w e r = 35 6 ⋅ 6 = 35 f(x) = \dfrac{x^3}{3} - \dfrac32x^2 + 2x + 5 \\ So, f'(x) = x^2 - 3x + 2 \\ {\text{For finding maximum and minimum value, we need to find the points whose value is at extrema. For finding points, }} \\ f'(x) = 0 \implies x^2 - 3x + 2 = 0 \implies (x - 2)(x - 1) = 0 \implies x = 1,2 \\ {\text{Now we will identify which point has maximum value, for that we will first find }} f''(x). \\ So, f''(x) = 2x - 3 \\ {\text{Putting }} x = 1,2 {\text{ in }} f''(x), {\text{we have}} \\ f''(1) = -1 < 0 .......................({\text{Since, value is negative so given function has maximum value at }} x = 1) \\ Now, f''(2) = 1 >0 ...................({\text{Since, value is positive so given function has minimum value at }} x = 2) \\ {\text{We need to find the maximum value of }} f(x), \\ So, {\text{Max. value}} = f(1) = \dfrac{1}{3} - \dfrac{3}{2} + 2 + 5 = \dfrac{35}{6}. \\ \therefore Answer = \dfrac{35}{6} \cdot 6 = \color{#EC7300}{\boxed{35}}

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Chew-Seong Cheong - 4 years ago

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