Problem 49 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(x \in[0,2]\) and \(g(x)=f(x)+f(2-x)\). Also, \(f^{\prime \prime}(x)<0\) then \(g(x)\) (A) increases in \([0,2]\) (B) decreases in \([0,2]\) (C) decreases in \([0,1)\) and increases in \((1,2]\) (D) increases in \([0,1)\) and decreases in \((1,2]\)

Step-by-Step Solution

Answer

(C) decreases in \([0,1)\) and increases in \((1,2]\)

Step 1: Understand Function g(x)

Given the function \(g(x) = f(x) + f(2-x)\), this implies \(g(x)\) is defined over the interval \([0,2]\). The function \(f(x)\) is combined with its symmetric counterpart over \([0,2]\).

Step 2: Assess f(x) Properties

The function \(f(x)\) is provided with \(f''(x) < 0\), indicating that \(f(x)\) is concave down over its domain. This means \(f(x)\) is a concave function, suggesting a unique turning point (maximum) within its domain, but no minima.

Step 3: Compute g'(x)

Differentiate \(g(x) = f(x) + f(2-x)\) with respect to \(x\) to find \(g'(x)\). Using the chain rule for the second term, we get: \[ g'(x) = f'(x) - f'(2-x) \].

Step 4: Evaluate at Turning Point

To determine where \(g(x)\) is increasing or decreasing, check changes in \(g'(x)\). Since \(f(x)\) is concave, and for symmetrically selected \(1\) as the midpoint in \([0,2]\), \(g'(1) = f'(1) - f'(1) = 0\), indicating a potential turning point.

Step 5: Analyze g(x) Behavior in [0,2]

In the interval \([0,1)\), \(f'(x) < 0\) due to concavity of \(f(x)\), meaning \(g'(x) = f'(x) - f'(2-x) < 0\) results in decrease. For \((1,2]\), \(-f'(2-x) > f'(x)\) since \(-f'(2-x)\) becomes more negative in symmetry, leading to \(g'(x) > 0\) indicating an increase.

Step 6: Conclude the Behavior of g(x)

The function \(g(x)\) subsequently decreases on \([0,1)\) and increases on \((1,2]\) based on the described derivative behavior. Hence, the correct response is option (C).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus, essential for understanding how a function behaves and changes. When we differentiate a function, we find its derivative, which tells us how the function's value changes as its input changes. In simpler terms, the derivative gives us the slope of the function at any given point.

For the given problem, we have a function expressed as a sum: \(g(x) = f(x) + f(2-x)\). To analyze this composite function, we find its derivative, \(g'(x)\), using differentiation rules. Differentiation of this function involves using the power of the chain rule. The chain rule helps us differentiate composite functions by differentiating the outer function and then multiplying it by the derivative of the inner function.

• For \(f(x)\), differentiate directly to get \(f'(x)\)
• For \(f(2-x)\), apply the chain rule: the derivative becomes \(-f'(2-x)\)

The resulting derivative \(g'(x) = f'(x) - f'(2-x)\) is key to determining the behavior of \(g(x)\). With this, we can analyze where \(g(x)\) increases or decreases over the given interval \([0, 2]\).

Function Behavior

Understanding a function's behavior involves determining where it increases or decreases. This is typically assessed using the derivative we computed. In the problem of \(g(x)\), we're interested in how and why \(g(x)\) changes over its domain.

Given that \(f''(x) < 0\), function \(f(x)\) is concave down, suggesting it has a downward curving shape. This property implies that the first derivative, \(f'(x)\), will be decreasing throughout. Now, when considering \(g'(x) = f'(x) - f'(2-x)\), observing its sign over the interval \([0,2]\) will reveal where \(g(x)\) increases or decreases.

• In the interval \([0, 1)\), \(f'(x)\) is expected to be less than zero, making \(g'(x) = f'(x) - f'(2-x) < 0\). This indicates a decreasing \(g(x)\).
• In the interval \((1, 2]\), the asymmetry and concavity cause \(-f'(2-x)\) to dominate over \(f'(x)\), resulting in \(g'(x) > 0\). Here, \(g(x)\) increases.

Thus, understanding these changes helps us conclude the behavior of \(g(x)\) across its domain.

Symmetry in Functions

Symmetry is an intriguing property that can greatly simplify the analysis of functions. For this problem, the symmetry arises in the form of the expression \(f(x) + f(2-x)\). This particular symmetry tells us that for every point \(x\) in \([0, 2]\), there is a corresponding point \(2-x\) to consider.

Interestingly, symmetry can reveal potential turning points or behaviors inherent in a function. In this exercise, symmetry plays a vital role in the computation and analysis of \(g'(x) = f'(x) - f'(2-x)\). It highlights that at \(x=1\), the derivatives \(f'(1)\) and \(-f'(1)\) cancel each other out, giving \(g'(1) = 0\). This point acts as a potential turning point due to the balance in symmetry.

• The symmetry ensures that any deviations on one side of \(x=1\) have mirrored effects on the other side.
• It allows for a quick comparison of function behavior without individually assessing each value.

This characteristic simplifies function behavior analysis by centering around key symmetrical points, thereby revealing unique insights into the nature of \(g(x)\).