Function Value Bound Using Derivative
We are given a continuous and differentiable function $f(x)$ on the interval $[3, 18]$. We know that $f(3) = -50$ and the derivative satisfies $f'(x) \le 20$ for all $x$ in $[3, 18]$. We need to find the largest possible value for $f(18)$.
Applying Derivative Inequality
Since $f(x)$ is continuous and differentiable, we can relate the change in the function's value to its derivative. The inequality $f'(x) \le 20$ implies that the slope of the function never exceeds 20.
We can use the property derived from the Mean Value Theorem, which states that for a function $f$ continuous on $[a, b]$ and differentiable on $(a, b)$, if $f'(x) \le M$ for all $x \in (a, b)$, then $f(b) - f(a) \le M(b-a)$.
In this case, $a=3$, $b=18$, and $M=20$. Applying this property:
$ f(18) - f(3) \le 20(18 - 3) $
Calculating Largest Possible Value
Substitute the known values into the inequality:
$ f(18) - (-50) \le 20(15) $
$ f(18) + 50 \le 300 $
Now, isolate $f(18)$ to find its maximum possible value:
$ f(18) \le 300 - 50 $
$ f(18) \le 250 $
Therefore, the largest possible value for $f(18)$ is 250. This value is achievable if $f'(x) = 20$ for all $x$ in $[3, 18]$.
Important Questions from Mean Value Theorem
A series expansion for the function sin θ is
If f is the derivative of some function on [a, b], then there exists a number c in (a, b) such that Integral of f with respect to x =
Which condition is not required in checking for Taylor's theorem?
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According to the Mean Value Theorem, for a continuous function f(x) in the interval [a, b], there exists a value ξ in this interval such that \(\mathop \smallint \limits_a^b f\left( x \right)dx =\)
