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Question 30 : Let f(x)= max{5x, 52 - 2x²}, where x is any positive real number.Then the minimum possible value of f(x) is [TITA]

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Explanatory Answer

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Method of solving this CAT Question from Functions

Given that f(x) = max{5x , 52 - 2x²}, where x is any positive real number.
The minimum possible value of f(x) has to found.
The minimum possible value is obtained if the two curves intersect or
5x = 52 - 2x²
2x² + 5x – 52 = 0
2x² – 8x + 13x – 52 = 0
2x(x - 4) + 13(x - 4) = 0
(2x + 13)(x - 4) = 0
x = \\frac{-13}{2}) or 4
One of these would be the minimum possible value
5x = 5 × \\frac{-13}{2}) = \\frac{-65}{2})
52 - 2x² = 52 – 2(\\frac{-13}{2}))² = \\frac{-65}{2})
Similarly when x = 4
5x = 5 × 4 =20
52 - 2x² = 52 – 2(4)² = 20
At x = 4 and \\frac{-13}{2}) the values are equal
And when x = \\frac{-13}{2}) , f(x) = \\frac{-65}{2})
The minimum possible value of f(x) is \\frac{-65}{2})