Question
Given that \( f(x)=x^{5} \). Find the value of \( x \) if \( f^{\prime}(x)=405 \)
Ask by Frank Estrada. in Malaysia
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = 3 \) or \( x = -3 \).
Solution
We are given \( f(x) = x^5 \) and need to find \( x \) such that
\[
f'(x) = 405.
\]
**Step 1: Compute the derivative**
Using the power rule, the derivative of \( f(x) = x^5 \) is
\[
f'(x) = 5x^4.
\]
**Step 2: Set up the equation**
Set the derivative equal to 405:
\[
5x^4 = 405.
\]
**Step 3: Solve for \( x^4 \)**
Divide both sides by 5:
\[
x^4 = \frac{405}{5} = 81.
\]
**Step 4: Find \( x \)**
We need to solve
\[
x^4 = 81.
\]
Taking the fourth root of both sides gives
\[
x = \pm \sqrt[4]{81}.
\]
Since \( 3^4 = 81 \), we have
\[
x = \pm 3.
\]
**Final Answer:**
\[
x = 3 \quad \text{or} \quad x = -3.
\]
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Simplify this solution The Deep Dive
To find the value of \( x \) such that \( f^{\prime}(x) = 405 \) for the function \( f(x) = x^5 \), we first need to calculate the derivative of \( f(x) \). The derivative \( f^{\prime}(x) \) is given by: \[ f^{\prime}(x) = 5x^4 \] Now, we set the derivative equal to 405: \[ 5x^4 = 405 \] Next, we solve for \( x^4 \): \[ x^4 = \frac{405}{5} = 81 \] Now, we take the fourth root of both sides to find \( x \): \[ x = \sqrt[4]{81} \] Since \( 81 = 3^4 \), we have: \[ x = \sqrt[4]{3^4} = 3 \] Thus, the value of \( x \) is: \[ \boxed{3} \]
