Let f(x)=√((1-x^2)(1+2 x^2)) defined on [-1,1] then (a) the greatest value of f(x) is I (b) the

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Let f(x)=√((1-x^2)(1+2 x^2)) defined on [-1,1] then (a) the...

Let $f(x)=\sqrt{\left(1-x^{2}\right)\left(1+2 x^{2}\right)}$ defined on $[-1,1]$ then
(a) the greatest value of $f(x)$ is I
(b) the greatest value of $f(x)$ is $3 \sqrt{8}$
(c) the least value of $f(x)$ is 0
(d) the least value of $f(x)$ is $-1$.

Key Concepts

Extreme Value Theorem

This theorem states that if a function is continuous on a closed interval, then it must attain both a maximum and a minimum value on that interval. This concept is central when analyzing functions defined on closed intervals such as [-1,1], ensuring that the greatest and least values exist.

Properties of the Square Root Function

The square root function inherently returns nonnegative values for nonnegative inputs. When a function is defined as the square root of another expression, its output is always greater than or equal to zero. This property immediately informs the determination of the least value of such functions.

Domain and Range Considerations

Understanding the domain and range is crucial when working with composite functions like the one in question. The domain defines the set of input values for which the function is defined, while the range is the set of outputs the function attains. For functions involving square roots, ensuring the radicand remains nonnegative is a key step in determining these sets.

Critical Point Analysis

Identifying critical points through differentiation is a standard method for finding local extrema of functions. In problems involving the determination of the greatest value of a function, analyzing where the derivative equals zero or does not exist helps locate potential maximums within the given interval.

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