Answer to Question #246166 in Real Analysis for tannu

Question

Determine the local minimum and local maximum values of the function f defined by f(X)=3-5x³+5x⁴-x⁵

Answer 1

f(x) = 3 – 5x³ + 5x⁴ – x⁵

To find the local minimum and maxima, we need to differentiate f(x) wrt x

f’(x) = –15x² + 20x³ – 5x⁴

Put f’(x) = 0

–15x² + 20x³ – 5x⁴ = 0

–5x² (x² – 4x + 3) = 0

x = 1, 3

So, these are the points of extrema. Lets find f”(x).

f”(x) = –30x + 60x –20x³

At x = 1, f”(x) > 0, so it is a point of minima.

At x = 3, f”(x) < 0, so it is a point of maxima.

Putting x = 1 in f(x), local minima value is 2

Putting x = 3 in f(x), local maxima value is 30.