(a) Show that the function f(x)=x^4-2 x^3 is not one-toone on (-∞,+∞). (b) Find the smallest val

(a) Show that the function f(x)=x^4-2 x^3 is not one-toone...

Key Concepts

Domain Restrictions to Achieve Injectivity

Even if a function is not one-to-one over its entire domain, it can be made injective by appropriately restricting its domain to an interval where it demonstrates monotonic behavior. This restriction is crucial for tasks such as finding inverse functions and ensuring that the function’s mapping from inputs to outputs remains one-to-one.

Critical Points and Their Role in Function Behavior

Critical points occur where a function's derivative is zero or undefined and are typically associated with local minima or maxima. Identifying these points is important as they highlight where the function changes its increasing or decreasing trends. This insight helps in determining subintervals over which the function remains monotonic and can therefore be restricted to become one-to-one.

Monotonicity and the Use of Derivatives

A function that is either entirely non-decreasing or non-increasing over an interval is known to be monotonic, and monotonic functions are inherently one-to-one on that interval. Derivatives provide a practical tool for analyzing monotonicity; if the derivative maintains a consistent sign (always positive or always negative) on an interval, the function does not change direction and is thus injective on that interval.

Injective (One-to-One) Functions

An injective function is one where every element of the domain corresponds to a unique element in the range. In other words, if two outputs are equal, the corresponding inputs must also be equal. This concept is essential when determining whether a function has an inverse, as only one-to-one functions have well-defined inverses.

Transcript

00:04 This question, you're given f of x equals x to the fourth minus 2x cubed.

00:10 You're asked to show that it is not 1 to 1 on the real numbers.

00:17 And then part b, you're asked to find an interval from k to infinity on which f of x is 1 to 1.

00:24 When you're talking about 1 to 1 functions, i know that if f prime of x is either always positive or always negative, then my function is one to one.

00:36 If i'm strictly increasing or strictly decreasing.

00:40 So i'm going to check f prime of x.

00:42 F prime of x is 4x cubed minus 6x squared, and in factored form i could rewrite it 2x squared times 2x minus 3.

00:58 Now if i look at a graph, a number line of f prime, and i label the two places where f prime is equal to 0, i can label each of the intervals around those zeros with positive and negative signs to say where is f prime positive and where is it negative.

01:17 2x squared is always a positive number or zero.

01:21 It's zero here, positive everywhere else.

01:25 So x equals three halves is the point where this f prime changes sign.

01:30 When x is larger than three halves, this factor is positive and so is this one.

01:37 When x is less than three halves, this factor is negative and this one's still positive.

01:46 And when x is less than zero, this one is positive, this one's still negative...

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